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"Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Time s" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Section heading" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT 302 55 "Introduction to the concept of t he limit of a function " }}{PARA 0 "" 0 "" {TEXT -1 63 "by Peter Stone , Mathematics Dept., Malaspina University-College" }}{PARA 0 "" 0 "" {TEXT -1 17 "Version: 5.9.2005" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 59 "Th e concept of the limit of a function and limit notation " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 224 " Many functions that we work with in mathematics have the property that , if input number is changed by a small amount, the ouput or value of \+ the function also changes by a small amount.\nFor example, with the fu nction f where " }{XPPEDIT 18 0 "f(x) = x^2;" "6#/-%\"fG6#%\"xG*$F'\" \"#" }{TEXT -1 11 ", we have " }{XPPEDIT 18 0 "f(2) = 4;" "6#/-%\"fG6 #\"\"#\"\"%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(2.01) = 4.0401;" "6 #/-%\"fG6#-%&FloatG6$\"$,#!\"#-F(6$\"&,/%!\"%" }{TEXT -1 36 ". In fac t, the closer that we take " }{TEXT 262 1 "x" }{TEXT -1 18 " to 2, the closer " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 7 " is to \+ " }{XPPEDIT 18 0 "f(2) = 4" "6#/-%\"fG6#\"\"#\"\"%" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "Digits := 11:\nf := x -> x^2: 'f(x)'=f(x);\n'f(2.1)'=f(2.1);\n'f( 2.01)'=f(2.01);\n'f(2.001)'=f(2.001);\n'f(2.0001)'=f(2.0001);\n'f(2.00 001)'=f(2.00001);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %\"fG6#%\"xG*$)F'\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"f G6#$\"#@!\"\"$\"$T%!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#$ \"$,#!\"#$\"&,/%!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#$\"%, ?!\"$$\"(,S+%!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#$\"&,+#! \"%$\"*,+/+%!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#$\"',+?! \"&$\",,+S++%!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "The seque nce of " }{TEXT 266 1 "x" }{TEXT -1 8 " values " }{XPPEDIT 18 0 "2.1, \+ 2.01, 2.001, 2.0001, 2.00001,` . . . `" "6(-%&FloatG6$\"#@!\"\"-F$6$\" $,#!\"#-F$6$\"%,?!\"$-F$6$\"&,+#!\"%-F$6$\"',+?!\"&%(~.~.~.~G" }{TEXT -1 85 " are decreasing towards 2. They produce the corresponding seque nce of output values: " }}{PARA 256 "" 0 "" {TEXT -1 15 "f(2.1) = 4.41 , " }}{PARA 256 "" 0 "" {TEXT -1 18 " f(2.01) = 4.0401," }}{PARA 256 " " 0 "" {TEXT -1 22 " f(2.001) = 4.004001, " }}{PARA 256 "" 0 "" {TEXT -1 23 "f(2.0001) = 4.00040001," }}{PARA 256 "" 0 "" {TEXT -1 26 "f(2.0 0001) = 4.0000400001," }}{PARA 256 "" 0 "" {TEXT -1 3 " : " }}{PARA 0 "" 0 "" {TEXT -1 113 "which are decreasing towards 4, that is, they ap proach 4 \"from above\" or from the right on the real number line. " } }{PARA 0 "" 0 "" {TEXT -1 25 "If we consider values of " }{TEXT 289 1 "x" }{TEXT -1 63 " close to 2, but less than 2, and approaching 2, the values of " }{XPPEDIT 18 0 "f(x)=x^2" "6#/-%\"fG6#%\"xG*$F'\"\"#" } {TEXT -1 18 " also approach 4. " }}{PARA 0 "" 0 "" {TEXT -1 23 "Taking the sequence of " }{TEXT 265 1 "x" }{TEXT -1 8 " values " }{XPPEDIT 18 0 "1.9, 1.99, 1.999, 1.9999,` . . . `" "6'-%&FloatG6$\"#>!\"\"-F$6$ \"$*>!\"#-F$6$\"%**>!\"$-F$6$\"&***>!\"%%(~.~.~.~G" }{TEXT -1 87 ", wh ich are increasing towards 2 produces the corresponding sequence of ou tput values: " }}{PARA 256 "" 0 "" {TEXT -1 16 " f(1.9) = 3.61, " }} {PARA 256 "" 0 "" {TEXT -1 19 " f(1.99) = 3.9601, " }}{PARA 256 "" 0 " " {TEXT -1 22 " f(1.999) = 3.996001, " }}{PARA 256 "" 0 "" {TEXT -1 24 " f(1.9999) = 3.99960001," }}{PARA 256 "" 0 "" {TEXT -1 28 " f(1.99 999) = 3.9999600001, " }}{PARA 256 "" 0 "" {TEXT -1 3 " : " }}{PARA 0 "" 0 "" {TEXT -1 112 "which are increasing towards 4, that is, they ap proach 4 \"from below\", or from the left on the real number line." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "Digits := 11:\nf := x -> x^2: 'f(x)'=f(x);\n'f(1.9)'=f(1.9);\n'f( 1.99)'=f(1.99);\n'f(1.999)'=f(1.999);\n'f(1.9999)'=f(1.9999);\n'f(1.99 999)'=f(1.99999);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %\"fG6#%\"xG*$)F'\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"f G6#$\"#>!\"\"$\"$h$!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#$ \"$*>!\"#$\"&,'R!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#$\"%* *>!\"$$\"(,g*R!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#$\"&*** >!\"%$\"*,+'**R!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#$\"'** **>!\"&$\",,+g***R!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "We use the arrow symbol to represent a variable approaching a number: " } {XPPEDIT 18 0 "x->2" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"#F*F*F *" }{TEXT -1 8 " means \"" }{TEXT 267 1 "x" }{TEXT -1 16 " approaches \+ 2\". " }}{PARA 0 "" 0 "" {TEXT -1 33 "Thus the fact that the values of " }{XPPEDIT 18 0 "f(x)=x^2" "6#/-%\"fG6#%\"xG*$F'\"\"#" }{TEXT -1 17 " approaches 4 as " }{TEXT 268 1 "x" }{TEXT -1 35 " approaches 2 can b e written: as " }{XPPEDIT 18 0 "x->2" "6#f*6#%\"xG7\"6$%)operatorG%& arrowG6\"\"\"#F*F*F*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "f(x)->4" "6#f*6 #-%\"fG6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"%F-F-F-" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 29 "We make use of the following " } {TEXT 259 14 "limit notation" }{TEXT -1 26 " to express the same fact. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(f(x),x=2)=4 " "6#/-%&LimitG6$-%\"fG6#%\"xG/F*\"\"#\"\"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 45 "When we use this notation we should consider " }{TEXT 275 1 "x" }{TEXT -1 15 " approaching 2 " }{TEXT 259 21 "from ei ther direction" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 100 "We can restrict attention to an approach from just one direction by modifying the notation slightly." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x ->2" "6#f*6#%\"xG7\"6$%)op eratorG%&arrowG6\"\"\"#F*F*F*" }{TEXT -1 9 "- means \"" }{TEXT 271 1 " x" }{TEXT -1 33 " approaches 2 from the left\" (so " }{TEXT 272 1 "x" }{TEXT -1 24 " increases towards 2). " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x ->2" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\" #F*F*F*" }{TEXT -1 9 "+ means \"" }{TEXT 269 1 "x" }{TEXT -1 34 " appr oaches 2 from the right\" (so " }{TEXT 270 1 "x" }{TEXT -1 22 " decrea ses towards 2)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The fact that " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 17 " approaches 4 as " }{TEXT 273 1 "x" }{TEXT -1 1 " " } {TEXT 259 9 "increases" }{TEXT -1 55 " towards 2 can be expressed in t he limit notation as: " }{XPPEDIT 18 0 "Limit(f(x),x = 2,left) = 4" " 6#/-%&LimitG6%-%\"fG6#%\"xG/F*\"\"#%%leftG\"\"%" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 14 "The fact that " }{XPPEDIT 18 0 "f(x)" "6# -%\"fG6#%\"xG" }{TEXT -1 17 " approaches 4 as " }{TEXT 274 1 "x" } {TEXT -1 1 " " }{TEXT 259 9 "decreases" }{TEXT -1 55 " towards 2 can b e expressed in the limit notation as: " }{XPPEDIT 18 0 "Limit(f(x),x \+ = 2,right) = 4" "6#/-%&LimitG6%-%\"fG6#%\"xG/F*\"\"#%&rightG\"\"%" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "For another example, consider the function " }{XPPEDIT 18 0 "g(x)=1/x" "6#/-%\"gG6#%\"xG*&\"\"\"F)F'!\"\"" }{TEXT -1 10 ". We have " }{XPPEDIT 18 0 "g(3)=1/3" "6#/-%\"gG6#\"\"$*&\"\"\"F)F'!\"\"" }{TEXT -1 1 " " }{TEXT 264 1 "~" }{TEXT -1 16 " 0.3333333333. " }} {PARA 0 "" 0 "" {TEXT -1 16 "The sequence of " }{TEXT 276 1 "x" } {TEXT -1 8 " values " }{XPPEDIT 18 0 "3.1,3.01.3.001,3.0001,3.00001,` \+ . . . `" "6'-%&FloatG6$\"#J!\"\"-%\".G6$-F$6$\"$,$!\"#-F$6$\"%,I!\"$-F $6$\"&,+$!\"%-F$6$\"',+I!\"&%(~.~.~.~G" }{TEXT -1 66 ", produces the c orresponding sequence of (approximate) values for " }{XPPEDIT 18 0 "g( x)=1/x" "6#/-%\"gG6#%\"xG*&\"\"\"F)F'!\"\"" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 25 " g(3.1) = 0.32258064516, " }}{PARA 256 "" 0 " " {TEXT -1 25 " g(3.01) =0.33222591362, " }}{PARA 256 "" 0 "" {TEXT -1 27 " g(3.001) = 0.33322225925, " }}{PARA 256 "" 0 "" {TEXT -1 28 " \+ g(3.0001) = 0.33332222259, " }}{PARA 256 "" 0 "" {TEXT -1 29 " g(3.000 01) = 0.33333222223, " }}{PARA 256 "" 0 "" {TEXT -1 3 " : " }}{PARA 0 "" 0 "" {TEXT -1 29 "These values are approaching " }{XPPEDIT 18 0 "1/ 3" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 50 " from below, that is, they a re increasing towards " }{XPPEDIT 18 0 "1/3" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 "g(x)->1/3" "6#f*6#-%\"gG6#%\"xG7\"6$%)operatorG%&arrowG6\"*&\"\"\"F/ \"\"$!\"\"F-F-F-" }{TEXT -1 6 "-, as " }{XPPEDIT 18 0 "x->3" "6#f*6#% \"xG7\"6$%)operatorG%&arrowG6\"\"\"$F*F*F*" }{TEXT -1 20 "+, so in par ticular " }{XPPEDIT 18 0 "g(x)->1/3" "6#f*6#-%\"gG6#%\"xG7\"6$%)operat orG%&arrowG6\"*&\"\"\"F/\"\"$!\"\"F-F-F-" }{TEXT -1 16 " (no minus), a s " }{XPPEDIT 18 0 "x->3" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"$ F*F*F*" }{TEXT -1 12 "+, that is, " }{XPPEDIT 18 0 "Limit(g(x),x=3,rig ht)=1/3" "6#/-%&LimitG6%-%\"gG6#%\"xG/F*\"\"$%&rightG*&\"\"\"F/F,!\"\" " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "Digits := 11:\ng := x -> 1/x: 'g(x)'=g(x);\n 'g(3)'=g(3);\n'g(3.1)'=g(3.1);\n'g(3.01)'=g(3.01);\n'g(3.001)'=g(3.001 );\n'g(3.0001)'=g(3.0001);\n'g(3.00001)'=g(3.00001);\nDigits := 10:" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG*&\"\"\"F)F'!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#\"\"$#\"\"\"F'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"gG6#$\"#J!\"\"$\",;X1eA$!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#$\"$,$!\"#$\",i8fAK$!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#$\"%,I!\"$$\",DfAAL$!#6" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"gG6#$\"&,+$!\"%$\",fAAKL$!#6" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"gG6#$\"',+I!\"&$\",BAALL$!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "The sequence of " }{TEXT 277 1 "x" }{TEXT -1 8 " values " }{XPPEDIT 18 0 "2.9,2.99,2.999,2.9999,2.99999,` . . . `" "6 (-%&FloatG6$\"#H!\"\"-F$6$\"$*H!\"#-F$6$\"%**H!\"$-F$6$\"&***H!\"%-F$6 $\"'****H!\"&%(~.~.~.~G" }{TEXT -1 63 ", produces the corresponding se quence of (approximate) values: " }}{PARA 256 "" 0 "" {TEXT -1 25 " g( 2.9) = 0.34482758621, " }}{PARA 256 "" 0 "" {TEXT -1 25 "g(2.99) = 0.3 3444816054, " }}{PARA 256 "" 0 "" {TEXT -1 26 "g(2.999) = 0.3334444814 9, " }}{PARA 256 "" 0 "" {TEXT -1 28 " g(2.9999) = 0.33334444481, " }} {PARA 256 "" 0 "" {TEXT -1 27 "g(2.99999) = .33333444445, " }}{PARA 256 "" 0 "" {TEXT -1 3 " : " }}{PARA 0 "" 0 "" {TEXT -1 29 "These valu es are approaching " }{XPPEDIT 18 0 "1/3" "6#*&\"\"\"F$\"\"$!\"\"" } {TEXT -1 50 " from above, that is, they are decreasing towards " } {XPPEDIT 18 0 "1/3" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 "g(x)->1/3" "6#f*6#-%\"gG 6#%\"xG7\"6$%)operatorG%&arrowG6\"*&\"\"\"F/\"\"$!\"\"F-F-F-" }{TEXT -1 6 "+, as " }{XPPEDIT 18 0 "x->3" "6#f*6#%\"xG7\"6$%)operatorG%&arro wG6\"\"\"$F*F*F*" }{TEXT -1 20 "-, so in particular " }{XPPEDIT 18 0 " g(x)->1/3" "6#f*6#-%\"gG6#%\"xG7\"6$%)operatorG%&arrowG6\"*&\"\"\"F/\" \"$!\"\"F-F-F-" }{TEXT -1 15 " (no plus), as " }{XPPEDIT 18 0 "x->3" " 6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"$F*F*F*" }{TEXT -1 12 "-, th at is, " }{XPPEDIT 18 0 "Limit(g(x),x = 3,left) = 1/3;" "6#/-%&LimitG6 %-%\"gG6#%\"xG/F*\"\"$%%leftG*&\"\"\"F/F,!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "Digits := 11:\ng := x -> 1/x: 'g(x)'=g(x);\n'g(2.9)'=g(2.9);\n'g( 2.99)'=g(2.99);\n'g(2.999)'=g(2.999);\n'g(2.9999)'=g(2.9999);\n'g(2.99 999)'=g(2.99999);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %\"gG6#%\"xG*&\"\"\"F)F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\" gG6#$\"#H!\"\"$\",@'eF[M!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG 6#$\"$*H!\"#$\",ag\"[WL!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6 #$\"%**H!\"$$\",\\\"[WML!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG 6#$\"&***H!\"%$\",\"[WWLL!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"g G6#$\"'****H!\"&$\",XWWLL$!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "In conclusion we have " }{XPPEDIT 18 0 "Limit(g(x),x=3)=1/3" "6#/- %&LimitG6$-%\"gG6#%\"xG/F*\"\"$*&\"\"\"F.F,!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 19 "For many functions " }{XPPEDIT 18 0 "f(x) " "6#-%\"fG6#%\"xG" }{TEXT -1 63 " it turns out that the value of the \+ limit of the function, say " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 5 ", as " }{TEXT 263 1 "x" }{TEXT -1 29 " approaches the give n number " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 19 ", say, i s just the " }{TEXT 259 21 "value of the function" }{TEXT -1 32 " at t hat input number, that is, " }{XPPEDIT 18 0 "Limit(f(x),x=a)=f(a)" "6# /-%&LimitG6$-%\"fG6#%\"xG/F*%\"aG-F(6#F," }{TEXT -1 110 ". This will b e the case if the function is continuous at a, which means, roughly sp eaking, that the values of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 28 " always change gradually as " }{TEXT 278 1 "x" }{TEXT -1 152 " changes gradually. This corresponds to the situation where the g raph is a continuous, although possibly jagged, line, with no break in the graph where " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT 260 7 "Warning" }{TEXT -1 27 ": This is not \+ always true. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The graph of the function " }{XPPEDIT 18 0 "f(x)=x^2" "6# /-%\"fG6#%\"xG*$F'\"\"#" }{TEXT -1 90 " is a smooth curve with no brea ks at all. In particular the graph passes through the point" } {XPPEDIT 18 0 "``(2,4)" "6#-%!G6$\"\"#\"\"%" }{TEXT -1 68 " with no br eak. This gives a pictorial explanation of the fact that " }{XPPEDIT 18 0 "Limit(f(x),x=2)" "6#-%&LimitG6$-%\"fG6#%\"xG/F)\"\"#" }{TEXT -1 38 " is just the value of the function at " }{XPPEDIT 18 0 "x=2" "6#/% \"xG\"\"#" }{TEXT -1 9 ", namely " }{XPPEDIT 18 0 "Limit(f(x),x=2)=f(2 )" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*\"\"#-F(6#F," }{XPPEDIT 18 0 "``=4" "6#/%!G\"\"%" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 381 370 370 {PLOTDATA 2 "69-%'CURVESG6$7S7$$\"\"!F)F(7$$\"3W mmmT&)G\\a!#>$\"3]iR.hXZpH!#?7$$\"3PL$ek`o!>5!#=$\"3gH$**>o+&Q5F-7$$\" 3omm\"z>)G_:F4$\"3ds8R\\')f4CF-7$$\"3-nmT&QU!*3#F4$\"3cet1)3)4kVF-7$$ \"3HL$eRZXKi#F4$\"3-\")Gk;oT\")oF-7$$\"3xm;z>,_=JF4$\"3^JaaPx;D(*F-7$$ \"3v**\\7G$[8j$F4$\"3,Bw,o!p'=8F47$$\"35n;z%*frhTF4$\"3az/8-!))>t\"F47 $$\"3A+]ilFQ!p%F4$\"3o-s!)[!p**>#F47$$\"3@ML$3_\"=M_F4$\"3J:LI>cmRFF47 $$\"3HnmTg(fJr&F4$\"3'HU/[W>SE$F47$$\"3k++]7eP_iF4$\"3c/N2I.A4RF47$$\" 3Q++]Pf!Qz'F4$\"3mD5k6*zbh%F47$$\"3@++](=ubJ(F4$\"3UG;GpDw^`F47$$\"37n ;zW(*Q*y(F4$\"3uflif#fu1'F47$$\"3#QLL3F-GN)F4$\"3;]Qcx0$p(pF47$$\"3=ML L$e'3I))F4$\"3/Oj\"p!H/(z(F47$$\"3?+]7.Rsim\\v*F47$$\"3;+DJ&H\"fT5!#<$\"3!>xqkU7\\3\"Fiq7$ $\"35+v$f)[$H4\"Fiq$\"3S;#**[m1X>\"Fiq7$$\"3cL$ek`1l9\"Fiq$\"30%oT\"Qs Z98Fiq7$$\"3OLe*[.-d>\"Fiq$\"3mOSDcLqH9Fiq7$$\"3km;/Egw[7Fiq$\"3a'*ez( e;%f:Fiq7$$\"3zm\"z%*f%)QI\"Fiq$\"39Z*e([]6+Fiq7$$\"3-++]i>Ad9Fi q$\"3e^B*z%e\\B@Fiq7$$\"32+]i:jf4:Fiq$\"3'\\d[h.\"))yAFiq7$$\"39+DJ&>r -c\"Fiq$\"3%y*=#H?YWV#Fiq7$$\"3++]P4q`;;Fiq$\"3()o!zE!>>8EFiq7$$\"3;LL $eM%4n;Fiq$\"3;p%4zb.#zFFiq7$$\"37++v$4v5s\"Fiq$\"3]rlKy%*4iHFiq7$$\"3 cm\"zWn*)*pFiq$\"3IR\"ymFiq$\"3^!3C$o**p6RFiq7$$\"3))*\\7G8O;.#Fiq$\"3$*H$\\hPX v7%Fiq7$$\"3!pmm;*\\[$3#Fiq$\"3V^-]5(44M%Fiq7$$\"3*pmT&Qz]O@Fiq$\"31j: XrhmkXFiq7$$\"3iLekG=4*=#Fiq$\"3!)fRCMI7#z%Fiq7$$\"3F++]i4TPAFiq$\"3Xx ^6:y+1]Fiq7$$\"3qL$3F9!z#H#Fiq$\"33,0]Qm)oD&Fiq7$$\"3'pmm;%>KUBFiq$\"3 [/JTy?Z'[&Fiq7$$\"3/+DJqJ8&R#Fiq$\"3727`.HmOdFiq7$$\"3G+voa-oXCFiq$\"3 k;$o\"3>N\")fFiq7$$\"3++++++++DFiq$\"3+++++++]iFiq-%'COLOURG6&%$RGBG$ \"*++++\"!\")F(F(-F$6%7%7$$\"3%**************p\"FiqF(7$Fc[l$\"37++++++ !*GFiq7$F(Ff[l-%&COLORG6&F[[l$\"\"%!\"\"F\\\\lF\\\\l-%*LINESTYLEG6#\" \"#-F$6%7%7$$\"3#)*************H#FiqF(7$Fg\\l$\"3/++++++!H&Fiq7$F(Fj\\ lFi[lF_\\l-F$6&7(Fb[lFe[lFh[lFf\\lFi\\lF\\]l-%'SYMBOLG6#%'CIRCLEG-Fiz6 &F[[lF)F)F)-%&STYLEG6#%&POINTG-F$6&F_]l-Fa]l6#%(DIAMONDGFd]lFf]l-F$6&F _]l-Fa]l6#%&CROSSGFd]lFf]l-F$6&7%7$$Fb\\lF)F(7$Fh^l$F]\\lF)7$F(Fj^l-Fa ]l6$Fc]l\"#:-Fiz6&F[[lF(F(F\\[lFf]l-F$6&Ff^l-Fa]l6$F^^lF^_lF__lFf]l-F$ 6&Ff^l-Fa]l6$Fc^lF^_lF__lFf]l-F$6$Ff^lF__l-F$6&7$7$$\"#Fb`lFa`l7%7$$\"+++]<>!\"*$!++++]8!#5Fc`l7$Fh`l$!++++]EF]al-Fg ]l6#%,PATCHNOGRIDGFd]l-F$6&7$7$$\"#BF^\\lFa`l7$$\"$0#Fb`lFa`l7%7$$\"++ +]#3#Fj`lF_alFjal7$F_blF[alFaalFd]l-F$6&7$7$$F^\\lF^\\l$\"$*GFb`l7$Ffb l$\"$!RFb`l7%7$$!++++]7F]al$\"++++*z$Fj`lFibl7$$!+++++v!#6F`clFaalFd]l -F$6&7$7$Ffbl$\"$H&Fb`l7$Ffbl$\"$5%Fb`l7%7$Fccl$\"++++>UFj`lF\\dl7$F^c lFadlFaalFd]l-%%TEXTG6$7$$\"#DF^\\lFa`lQ\"x6\"-Fedl6$7$Ffbl$\"#iF^\\lQ \"yF[el-Fedl6%7$Fh^lFa`lQ\"2F[elF__l-Fedl6%7$$Fb\\lF^\\l$\"#VF^\\lQ)f( 2)~=~4F[elF__l-Fedl6%7$Fd`l$\"$N%Fb`lQ&(2,4)F[elF__l-%*AXESTICKSG6$F)F )-%+AXESLABELSG6%FjdlQ!F[el-%%FONTG6#%(DEFAULTG-%*AXESSTYLEG6#%'NORMAL G-%%VIEWG6$;Fa`lFhdl;$!\"$F^\\l$\"$D'Fb`l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17 " "Curve 18" "Curve 19" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "In the picture the horizontal arrows below the " }{TEXT 281 1 "x" }{TEXT -1 40 " axis are supposed to sugg est values of " }{TEXT 282 1 "x" }{TEXT -1 25 " approaching 2 along th e " }{TEXT 280 1 "x" }{TEXT -1 74 " axis from either direction, while \+ the vertical arrows to the left of the " }{TEXT 279 1 "y" }{TEXT -1 48 " axis are supposed to suggest the corresponding " }{TEXT 283 1 "y " }{TEXT -1 20 " values approaching " }{XPPEDIT 18 0 "f(2)=4" "6#/-%\" fG6#\"\"#\"\"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 "The fo llowing animation illustrates this idea. " }}{PARA 0 "" 0 "" {TEXT -1 111 "To play the animation click on the graphic and use the controls i n the context bar, or use the animation menu. 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In particular the graph passes through the point" }{XPPEDIT 18 0 "``( 3, 1/3);" "6#-%!G6$\"\"$*&\"\"\"F(F&!\"\"" }{TEXT -1 68 " with no brea k. This gives a pictorial explanation of the fact that " }{XPPEDIT 18 0 "Limit(g(x),x = 3);" "6#-%&LimitG6$-%\"gG6#%\"xG/F)\"\"$" }{TEXT -1 38 " is just the value of the function at " }{XPPEDIT 18 0 "x = 3;" "6 #/%\"xG\"\"$" }{TEXT -1 9 ", namely " }{XPPEDIT 18 0 "Limit(g(x),x = 3 ) = g(2);" "6#/-%&LimitG6$-%\"gG6#%\"xG/F*\"\"$-F(6#\"\"#" }{XPPEDIT 18 0 "`` = 1/3;" "6#/%!G*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 2 ". 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JF1F]\\l7$F`^p$\"3=+++vjP@KF.7$F]\\lFc^pF]`lF_^l-F(6&Fd]pFi\\lF]_lF_]l -F(6&Fd]pFe]lF]_lF_]l-F(6&Fd]pFj]lF]_lF_]l-F(6&F^^pFi\\lF]`lF_]l-F(6&F ^^pFe]lF]`lF_]l-F(6&F^^pFj]lF]`lF_]l-F_al6%7$Fbal$\"+yCN`PFfalQ%.345Fh alF]_l-F_al6%7$$\"+Uot&p#F^blF_blQ%2.90FhalF]_l-F_al6%7$Fbal$\"+vjP@HF falQ%.323FhalF]`l-F_al6%7$$\"+eJE/LF^blF_blQ%3.10FhalF]`lF_clFhcl76F'F _\\lFc]lFh]lF_\\lF]^l-F(6%7%7$$\"33+++@%o$[HF1F]\\l7$F_ap$\"35+++&Q1fAAL$F.7$F]\\l F[epF]_lF_^l-F(6%7%7$$\"35++++++**HF1F]\\l7$Fbep$\"3,+++:[WMLF.7$F]\\l FeepF]`lF_^l-F(6&FfdpFi\\lF]_lF_]l-F(6&FfdpFe]lF]_lF_]l-F(6&FfdpFj]lF] _lF_]l-F(6&F`epFi\\lF]`lF_]l-F(6&F`epFe]lF]`lF_]l-F(6&F`epFj]lF]`lF_]l -F_al6%7$Fbal$\"+#fAAj$FfalQ%.333FhalF]_l-F_al6%7$$\"%,G!\"$F_blQ%3.00 FhalF]_l-F_al6%7$Fbal$\"+:[WMIFfalFifpF]`l-F_al6%7$$\"%*>$F_gpF_blF`gp F]`lF_clFhcl-%*AXESTICKSG6$F^\\lF^\\l-F`cl6$%\"xG%\"yG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve \+ 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve \+ 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" }}{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "First non- trivial example of a limit " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)=(x^3-8 )/(x-2)" "6#/-%\"fG6#%\"xG*&,&*$F'\"\"$\"\"\"\"\")!\"\"F,,&F'F,\"\"#F. F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 64 "We can construct a table of values of this function as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matri x([[x*`..`, 0., .2, .4, .6, .8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.4 , 2.6, 2.8, 3.0], [f(x)*`..`, 4.0, 4.44, 4.96, 5.56, 6.24, 7.00, 7.84, 8.76, 9.76, 10.84, undefined, 13.24, 14.55, 15.97, 17.44, 19.0]]);" " 6#-%'matrixG6#7$73*&%\"xG\"\"\"%#..GF*-%&FloatG6$\"\"!F/-F-6$\"\"#!\" \"-F-6$\"\"%F3-F-6$\"\"'F3-F-6$\"\")F3-F-6$\"#5F3-F-6$\"#7F3-F-6$\"#9F 3-F-6$\"#;F3-F-6$\"#=F3-F-6$\"#?F3-F-6$\"#AF3-F-6$\"#CF3-F-6$\"#EF3-F- 6$\"#GF3-F-6$\"#IF373*&-%\"fG6#F)F*F+F*-F-6$\"#SF3-F-6$\"$W%!\"#-F-6$ \"$'\\Fco-F-6$\"$c&Fco-F-6$\"$C'Fco-F-6$\"$+(Fco-F-6$\"$%yFco-F-6$\"$w )Fco-F-6$\"$w*Fco-F-6$\"%%3\"Fco%*undefinedG-F-6$\"%C8Fco-F-6$\"%b9Fco -F-6$\"%(f\"Fco-F-6$\"%WF3" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 9 " " } {XPPEDIT 18 0 "matrix([[x, f(x)], [_, _], [0., 4.00], [.2, 4.44], [.4, 4.96], [.6, 5.56], [.8, 6.24], [1.0, 7.00], [1.2, 7.84], [1.4, 8.76]] );" "6#-%'matrixG6#7,7$%\"xG-%\"fG6#F(7$%\"_GF-7$-%&FloatG6$\"\"!F2-F0 6$\"$+%!\"#7$-F06$\"\"#!\"\"-F06$\"$W%F67$-F06$\"\"%F;-F06$\"$'\\F67$- F06$\"\"'F;-F06$\"$c&F67$-F06$\"\")F;-F06$\"$C'F67$-F06$\"#5F;-F06$\"$ +(F67$-F06$\"#7F;-F06$\"$%yF67$-F06$\"#9F;-F06$\"$w)F6" }{TEXT -1 24 " " }{XPPEDIT 18 0 "matrix([[x, f(x)], [_, _], [ 1.6, 9.76], [1.8, 10.84], [2.0, undefined], [2.2, 13.24], [2.4, 14.55] , [2.6, 15.97], [2.8, 17.44], [3.0, 19.00]]);" "6#-%'matrixG6#7,7$%\"x G-%\"fG6#F(7$%\"_GF-7$-%&FloatG6$\"#;!\"\"-F06$\"$w*!\"#7$-F06$\"#=F3- F06$\"%%3\"F77$-F06$\"#?F3%*undefinedG7$-F06$\"#AF3-F06$\"%C8F77$-F06$ \"#CF3-F06$\"%b9F77$-F06$\"#EF3-F06$\"%(f\"F77$-F06$\"#GF3-F06$\"%WF7" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Notice that " }{XPPEDIT 18 0 "f(x) " "6#-%\"fG6#%\"xG" }{TEXT -1 21 " is not defined when " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 42 ". This happens because the denom inator of " }{XPPEDIT 18 0 "(x^3-8)/(x-2)" "6#*&,&*$%\"xG\"\"$\"\"\"\" \")!\"\"F(,&F&F(\"\"#F*F*" }{TEXT -1 14 " is zero when " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 63 ", and division by zero is imposs ible. Notice that the numerator" }{XPPEDIT 18 0 "``(x^3-8)" "6#-%!G6#, &*$%\"xG\"\"$\"\"\"\"\")!\"\"" }{TEXT -1 19 " is also zero when " } {XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 3 ". " }}{PARA 0 "" 0 " " {TEXT -1 36 "The points appear to lie on a curve." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "h := 0.2: \npts:=[seq([h*i,evalf(f(h*i),4)],i=0..15)]:\nplot(pts,style=point,sym bol=circle,color=black,view=[0..3,0..19]);" }}{PARA 13 "" 1 "" {GLPLOT2D 328 359 359 {PLOTDATA 2 "6(-%'CURVESG6#727$$\"\"!F)$\"\"%F)7 $$\"35+++++++?!#=$\"3R++++++SW!#<7$$\"3A+++++++SF/$\"3'*************f \\F27$$\"3w**************fF/$\"3g************fbF27$$\"3U+++++++!)F/$\" 3A++++++SiF27$$\"\"\"F)$\"\"(F)7$$\"3%**************>\"F2$\"3&)******* *****RyF27$$\"3!**************R\"F2$\"3!)************f()F27$$\"33+++++ ++;F2$\"3!)************f(*F27$$\"3/+++++++=F2$\"3+++++++%3\"!#;7$%*und efinedGFgn7$$\"3;+++++++AF2$\"3+++++++D8Fen7$$\"3!**************R#F2$ \"33++++++b9Fen7$$\"33+++++++EF2$\"31++++++(f\"Fen7$$\"3#)************ *z#F2$\"38++++++WF)-%+AXESLABELSG6$Q!6\"Fdp-%&STYLE G6#%&POINTG-%'SYMBOLG6#%'CIRCLEG-%'COLOURG6&%$RGBGF)F)F)-%%VIEWG6$;F(F ]p;F(F_p" 1 5 4 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur ve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The following table gi ves approximate values for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 20 " for some values of " }{TEXT 290 1 "x" }{TEXT -1 28 " in \+ the neighbourhood of 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[x*`..`, 1.70, 1.75, 1. 80, 1.85, 1.90, 1.95, 2., 2.05, 2.10, 2.15, 2.20, 2.25, 2.30], [f(x)*` ..`, 10.29, 10.5625, 10.84, 11.1225, 11.41, 11.7024, `?`, 12.3024, 12. 61, 12.9225, 13.24, 13.5624, 13.89]]);" "6#-%'matrixG6#7$70*&%\"xG\"\" \"%#..GF*-%&FloatG6$\"$q\"!\"#-F-6$\"$v\"F0-F-6$\"$!=F0-F-6$\"$&=F0-F- 6$\"$!>F0-F-6$\"$&>F0-F-6$\"\"#\"\"!-F-6$\"$0#F0-F-6$\"$5#F0-F-6$\"$:# F0-F-6$\"$?#F0-F-6$\"$D#F0-F-6$\"$I#F070*&-%\"fG6#F)F*F+F*-F-6$\"%H5F0 -F-6$\"'Dc5!\"%-F-6$\"%%3\"F0-F-6$\"'D76F[o-F-6$\"%T6F0-F-6$\"'Cq6F[o% \"?G-F-6$\"'CI7F[o-F-6$\"%h7F0-F-6$\"'D#H\"F[o-F-6$\"%C8F0-F-6$\"'Cc8F [o-F-6$\"%*Q\"F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "matrix([[x, f(x)], [ _, _],[1.65, 10.0225], [1.70, 10.2900], [1.75, 10.5625], [1.80, 10.840 0], [1.85, 11.1225], [1.90, 11.4100], [1.95, 11.7024]])" "6#-%'matrixG 6#7+7$%\"xG-%\"fG6#F(7$%\"_GF-7$-%&FloatG6$\"$l\"!\"#-F06$\"'D-5!\"%7$ -F06$\"$q\"F3-F06$\"'+H5F77$-F06$\"$v\"F3-F06$\"'Dc5F77$-F06$\"$!=F3-F 06$\"'+%3\"F77$-F06$\"$&=F3-F06$\"'D76F77$-F06$\"$!>F3-F06$\"'+T6F77$- F06$\"$&>F3-F06$\"'Cq6F7" }{TEXT -1 29 " \+ " }{XPPEDIT 18 0 "matrix([[x, f(x)], [_, _],[2., undefined], [2.05, 12 .3024], [2.10, 12.6100], [2.15, 12.9225], [2.20, 13.2400], [2.25, 13.5 624], [2.30, 13.8900]])" "6#-%'matrixG6#7+7$%\"xG-%\"fG6#F(7$%\"_GF-7$ -%&FloatG6$\"\"#\"\"!%*undefinedG7$-F06$\"$0#!\"#-F06$\"'CI7!\"%7$-F06 $\"$5#F9-F06$\"'+h7F=7$-F06$\"$:#F9-F06$\"'D#H\"F=7$-F06$\"$?#F9-F06$ \"'+C8F=7$-F06$\"$D#F9-F06$\"'Cc8F=7$-F06$\"$I#F9-F06$\"'+*Q\"F=" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 70 "The next picture shows the new points along with the prev ious points. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "h := 0.05:\npts2:=[seq([2+h*i,evalf(f(2+h*i),6) ],i=-7..6)]:\nplot([op(pts),op(pts2)],style=point,symbol=circle,\n \+ color=black,view=[0..3,0..19]);" }}{PARA 13 "" 1 "" {GLPLOT2D 383 367 367 {PLOTDATA 2 "6(-%'CURVESG6#7@7$$\"\"!F)$\"\"%F)7$$\"35++++ +++?!#=$\"3R++++++SW!#<7$$\"3A+++++++SF/$\"3'*************f\\F27$$\"3w **************fF/$\"3g************fbF27$$\"3U+++++++!)F/$\"3A++++++SiF 27$$\"\"\"F)$\"\"(F)7$$\"3%**************>\"F2$\"3&)************RyF27$ $\"3!**************R\"F2$\"3!)************f()F27$$\"33+++++++;F2$\"3!) ************f(*F27$$\"3/+++++++=F2$\"3+++++++%3\"!#;7$%*undefinedGFgn7 $$\"3;+++++++AF2$\"3+++++++D8Fen7$$\"3!**************R#F2$\"33++++++b9 Fen7$$\"33+++++++EF2$\"31++++++(f\"Fen7$$\"3#)*************z#F2$\"38++ ++++WF)7$$\"3#*************\\;F2$\"34+++++D-5Fen7$$ \"3%**************p\"F2$\"3#*************G5Fen7$$\"3+++++++]F2$\"33+++++Cq6FenFfn7$$\"3#)*** *********\\?F2$\"30+++++CI7Fen7$$\"33+++++++@F2$\"3%************4E\"Fe n7$$\"3#*************\\@F2$\"3%**********\\AH\"Fen7$Fin$\"3-++++++C8Fe n7$$\"3+++++++]AF2$\"3-+++++Cc8Fen7$$\"3#)*************H#F2$\"31++++++ *Q\"Fen-%+AXESLABELSG6$Q!6\"F^t-%&STYLEG6#%&POINTG-%'SYMBOLG6#%'CIRCLE G-%'COLOURG6&%$RGBGF)F)F)-%%VIEWG6$;F(F]p;F(F_p" 1 5 4 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Taking more values of " }{TEXT 291 1 "x" }{TEXT -1 68 " close to 2 fills out a curve with the exception of the point wit h " }{TEXT 297 1 "x" }{TEXT -1 15 " coordinate 2. " }}{PARA 0 "" 0 "" {TEXT -1 26 "The graph of the function " }{XPPEDIT 18 0 "f(x)=(x^3-8)/ (x-2)" "6#/-%\"fG6#%\"xG*&,&*$F'\"\"$\"\"\"\"\")!\"\"F,,&F'F,\"\"#F.F. " }{TEXT -1 29 " consists of a curve with a \"" }{TEXT 259 13 "missing point" }{TEXT -1 8 "\" where " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" } {TEXT -1 69 ", that is, a curve with a gap or hole, consisting of a si ngle point. " }}{PARA 0 "" 0 "" {TEXT -1 12 "What is the " }{TEXT 292 1 "y" }{TEXT -1 34 " coordinate of the missing point? " }}{PARA 0 "" 0 "" {TEXT -1 13 "To find this " }{TEXT 293 1 "y" }{TEXT -1 49 " coord inate we need to take find the value which " }{XPPEDIT 18 0 "f(x)" "6# -%\"fG6#%\"xG" }{TEXT -1 15 " approaches as " }{TEXT 294 1 "x" }{TEXT -1 37 " approaches 2 from either direction. " }}{PARA 0 "" 0 "" {TEXT -1 41 "This is precisely the value of the limit " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(f(x),x=2)=Limit((x^3-8)/(x-2),x=2 )" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*\"\"#-F%6$*&,&*$F*\"\"$\"\"\"\"\")! \"\"F3,&F*F3F,F5F5/F*F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "The value of this limit cannot be \+ the value of the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 4 " at " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 10 ", because " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 " " } {TEXT 259 12 "has no value" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "Digits := 16:\nf := x -> (x ^3-8)/(x-2): 'f(x)'=f(x);\n'f(2.1)'=f(2.1);\n'f(2.01)'=f(2.01);\n'f(2. 001)'=f(2.001);\n'f(2.0001)'=f(2.0001);\n'f(2.00001)'=f(2.00001);\nDig its := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&,&*$)F' \"\"$\"\"\"F-\"\")!\"\"F-,&F'F-\"\"#F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#$\"#@!\"\"$\"1++++++h7!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#$\"$,#!\"#$\"1+++++,17!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#$\"%,?!\"$$\"1++++,g+7!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#$\"&,+#!\"%$\"1+++,+1+7!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#$\"',+?!\"&$\"1++,+g++7!#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "The sequence of " }{TEXT 295 1 "x" }{TEXT -1 8 " values " }{XPPEDIT 18 0 "2.1, 2.01, 2.001, 2.0001, 2.00001,` . . . ` " "6(-%&FloatG6$\"#@!\"\"-F$6$\"$,#!\"#-F$6$\"%,?!\"$-F$6$\"&,+#!\"%-F $6$\"',+?!\"&%(~.~.~.~G" }{TEXT -1 83 ", which are decreasing towards \+ 2, produce the corresponding sequence of values for " }{XPPEDIT 18 0 " f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 16 " f(2.1) = 12.61 " }}{PARA 256 "" 0 "" {TEXT -1 19 " f(2.01) = 12.0 601 " }}{PARA 256 "" 0 "" {TEXT -1 22 " f(2.001) = 12.006001 " }} {PARA 256 "" 0 "" {TEXT -1 25 " f(2.0001) = 12.00060001 " }}{PARA 256 "" 0 "" {TEXT -1 28 " f(2.00001) = 12.0000600001 " }}{PARA 256 "" 0 " " {TEXT -1 3 " : " }}{PARA 0 "" 0 "" {TEXT -1 32 "which are decreasing towards 12." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "The sequence of " }{TEXT 296 1 "x" }{TEXT -1 8 " values " } {XPPEDIT 18 0 "1.9, 1.99, 1.999, 1.9999,` . . . `" "6'-%&FloatG6$\"#>! \"\"-F$6$\"$*>!\"#-F$6$\"%**>!\"$-F$6$\"&***>!\"%%(~.~.~.~G" }{TEXT -1 83 ", which are increasing towards 2, produce the corresponding seq uence of values for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 16 " f(1.9) = 11.41 " }}{PARA 256 "" 0 "" {TEXT -1 19 " f(1.99) = 11.9401 " }}{PARA 256 "" 0 "" {TEXT -1 22 " f(1.999) = 11.994001 " }}{PARA 256 "" 0 "" {TEXT -1 25 " f(1.9999) = 11.99940001 " }}{PARA 256 "" 0 "" {TEXT -1 28 " f(1.99999 ) = 11.9999400001 " }}{PARA 256 "" 0 "" {TEXT -1 3 " : " }}{PARA 0 "" 0 "" {TEXT -1 33 "which are increasing towards 12. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "These calculations sugg est that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(f( x),x=2)=Limit((x^3-8)/(x-2),x=2)" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*\"\"# -F%6$*&,&*$F*\"\"$\"\"\"\"\")!\"\"F3,&F*F3F,F5F5/F*F," }{XPPEDIT 18 0 "``=12" "6#/%!G\"#7" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "Digits := 16:\nf := x -> \+ (x^3-8)/(x-2): 'f(x)'=f(x);\n'f(1.9)'=f(1.9);\n'f(1.99)'=f(1.99);\n'f( 1.999)'=f(1.999);\n'f(1.9999)'=f(1.9999);\n'f(1.99999)'=f(1.99999);\nD igits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&,&*$)F '\"\"$\"\"\"F-\"\")!\"\"F-,&F'F-\"\"#F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#$\"#>!\"\"$\"1++++++T6!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#$\"$*>!\"#$\"1+++++,%>\"!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"fG6#$\"%**>!\"$$\"1++++,S*>\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#$\"&***>!\"%$\"1+++,+%**>\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#$\"'****>!\"&$\"1++,+S***>\"!#9" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "We can r epresent this situation by drawing the curve with the " }{TEXT 259 13 "missing point" }{XPPEDIT 18 0 "``(2,12)" "6#-%!G6$\"\"#\"#7" }{TEXT -1 13 " shown as an " }{TEXT 259 11 "open circle" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "f := x -> (x^3-8)/(x-2):\n'f(x)'=f(x);\nplot([f(x),[[2,12]]$2],x= -2..4,y=0..f(4),style=[line,point$2],\n symbol=circle,symbolsize =[15,18],color=[red,brown$2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"fG6#%\"xG*&,&*$)F'\"\"$\"\"\"F-\"\")!\"\"F-,&F'F-\"\"#F/F/" }}{PARA 13 "" 1 "" {GLPLOT2D 428 346 346 {PLOTDATA 2 "6(-%'CURVESG6&7S7$$!\"# \"\"!$\"\"%F*7$$!3!******\\2<#p=!#<$\"3Ob:ZB$Qbv$F07$$!31++D^NUbiUCFY$\"3[\"FY$\"3q3QQa^l_UF07$$\"3h** **\\(3wY_#FY$\"35Vz'46v'oXF07$$\"3F)******HOTq$FY$\"3iw(*G)*\\FY$\"3!\\kr*etX\\_F07$$\"3_,+]isVIiFY$\"3.&[>t$4FMcF07$$ \"3&=++](o:;vFY$\"3g)4'p;v:ogF07$$\"3#>++v$)[op)FY$\"3y^K_k:s&\\'F07$$ \"3W*****\\i%Qq**FY$\"3YjVq?E;))pF07$$\"3&****\\(QIKH6F0$\"3XL:N.m,MvF 07$$\"3#****\\7:xWC\"F0$\"3r9L[#oxw.)F07$$\"37++]Zn%)o8F0$\"3\")G898\\ V6')F07$$\"3y******4FL(\\\"F0$\"3(G%fVk1nO#*F07$$\"3#)****\\d6.B;F0$\" 3?*y:KX#H!))*F07$$\"3(****\\(o3lW\" H+>F`u7$$\"30++]_!>w7$F0$\"39+hD/Rs.?F`u7$$\"3O++v)Q?QD$F0$\"3D?X(**y) \\4@F`u7$$\"3G+++5jypLF0$\"3+Uj]RK]4AF`u7$$\"3<++]Ujp-NF0$\"3o(Gw_VFuK #F`u7$$\"3++++gEd@OF0$\"3eZf;&Q$*eV#F`u7$$\"39++v3'>$[PF0$\"3CvRo5RlaD F`u7$$\"37++D6EjpQF0$\"3[\\\\&o " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 46 "It is possible to find the va lue of the limit " }{XPPEDIT 18 0 "Limit((x^3-8)/(x-2),x = 2)" "6#-%&L imitG6$*&,&*$%\"xG\"\"$\"\"\"\"\")!\"\"F+,&F)F+\"\"#F-F-/F)F/" }{TEXT -1 53 " wthout resorting to lengthy numerical calculations. " }}{PARA 0 "" 0 "" {TEXT -1 54 "This involves an algebraic analysis of the expr ession " }{XPPEDIT 18 0 "(x^3-8)/(x-2)" "6#*&,&*$%\"xG\"\"$\"\"\"\"\") !\"\"F(,&F&F(\"\"#F*F*" }{TEXT -1 133 ". The main idea which leads to \+ the alternative shorter argument for the evaluation of our limit is a \+ \"factorisation\" of the numerator" }{XPPEDIT 18 0 "``(x^3-8);" "6#-%! G6#,&*$%\"xG\"\"$\"\"\"\"\")!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 64 "From experience with solving polynomial equations, the fa ct that" }{XPPEDIT 18 0 " ``(x^3-8)" "6#-%!G6#,&*$%\"xG\"\"$\"\"\"\"\" )!\"\"" }{TEXT -1 23 " is equal to zero when " }{XPPEDIT 18 0 "x=2" "6 #/%\"xG\"\"#" }{TEXT -1 14 " suggests that" }{XPPEDIT 18 0 "``(x-2)" " 6#-%!G6#,&%\"xG\"\"\"\"\"#!\"\"" }{TEXT -1 16 " is a factor of " } {XPPEDIT 18 0 "``(x^3-8)" "6#-%!G6#,&*$%\"xG\"\"$\"\"\"\"\")!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 129 "There are various meth ods which can be used to find the other factor. The most direct method is to perform polynomial division of" }{XPPEDIT 18 0 " ``(x^3-8)" "6# -%!G6#,&*$%\"xG\"\"$\"\"\"\"\")!\"\"" }{TEXT -1 3 " by" }{XPPEDIT 18 0 " ``(x-2)" "6#-%!G6#,&%\"xG\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "The comp leted division looks as follows, and proceeds in a similar manner to t he long division of integers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 28 " " } {XPPEDIT 18 0 "x^2+2*x+4" "6#,(*$%\"xG\"\"#\"\"\"*&F&F'F%F'F'\"\"%F'" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 31 " _______ __________ " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x-2" "6#,&%\"xG\"\"\"\"\"#!\"\"" }{TEXT -1 7 " | " }{XPPEDIT 18 0 "x^3+ 0*x^2+0*x-8" "6#,**$%\"xG\"\"$\"\"\"*&\"\"!F'*$F%\"\"#F'F'*&F)F'F%F'F' \"\")!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x^3-2*x^2" "6#,&*$%\"xG\"\"$\"\"\"*&\"\"#F'*$F%F)F'!\" \"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 13 " ________ " }} {PARA 256 "" 0 "" {TEXT -1 22 " " }{XPPEDIT 18 0 "2*x^2+0*x" "6#,&*&\"\"#\"\"\"*$%\"xGF%F&F&*&\"\"!F&F(F&F&" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 20 " " } {XPPEDIT 18 0 "2*x^2-4*x" "6#,&*&\"\"#\"\"\"*$%\"xGF%F&F&*&\"\"%F&F(F& !\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 28 " \+ ________ " }}{PARA 256 "" 0 "" {TEXT -1 35 " \+ " }{XPPEDIT 18 0 "4*x-8" "6#,&*&\"\"%\"\"\"%\"xGF&F&\"\" )!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 35 " \+ " }{XPPEDIT 18 0 "4*x-8" "6#,&*&\"\"%\"\"\"%\"xG F&F&\"\")!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 44 " \+ ______ " }}{PARA 256 "" 0 "" {TEXT -1 46 " 0 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The individual terms of t he dividend" }{XPPEDIT 18 0 "``(x^2+2*x+4);" "6#-%!G6#,(*$%\"xG\"\"#\" \"\"*&F)F*F(F*F*\"\"%F*" }{TEXT -1 65 " are obtained in three steps by dividing the highest degree term " }{TEXT 298 1 "x" }{TEXT -1 3 " of " }{XPPEDIT 18 0 " ``(x-2)" "6#-%!G6#,&%\"xG\"\"\"\"\"#!\"\"" }{TEXT -1 19 " successively into " }{XPPEDIT 18 0 "x^3, 2*x^2" "6$*$%\"xG\"\" $*&\"\"#\"\"\"*$F$F'F(" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "4*x" "6#* &\"\"%\"\"\"%\"xGF%" }{TEXT -1 17 ". The expressions" }{XPPEDIT 18 0 " ``(x^3-2*x^2),``(2*x^2-4*x);" "6$-%!G6#,&*$%\"xG\"\"$\"\"\"*&\"\"#F**$ F(F,F*!\"\"-F$6#,&*&F,F**$F(F,F*F**&\"\"%F*F(F*F." }{TEXT -1 4 " and" }{XPPEDIT 18 0 "``(4*x-8);" "6#-%!G6#,&*&\"\"%\"\"\"%\"xGF)F)\"\")!\" \"" }{TEXT -1 28 " are obtained by multiplying" }{XPPEDIT 18 0 " ``(x- 2)" "6#-%!G6#,&%\"xG\"\"\"\"\"#!\"\"" }{TEXT -1 22 " by each of the te rms " }{XPPEDIT 18 0 "x^2,2*x;" "6$*$%\"xG\"\"#*&F%\"\"\"F$F'" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "4;" "6#\"\"%" }{TEXT -1 136 " in the div idend as soon as the term is obtained. These expressions are used to o btain (by subtraction) the remainder in the division of" }{XPPEDIT 18 0 " ``(x^3-8)" "6#-%!G6#,&*$%\"xG\"\"$\"\"\"\"\")!\"\"" }{TEXT -1 68 " by the (partial) dividend constructed at that stage. The expression" }{XPPEDIT 18 0 "``(2*x^2+0*x)" "6#-%!G6#,&*&\"\"#\"\"\"*$%\"xGF(F)F)*& \"\"!F)F+F)F)" }{TEXT -1 34 " constitutes part of the remainder" } {XPPEDIT 18 0 "``(2*x^2+0*x-8)" "6#-%!G6#,(*&\"\"#\"\"\"*$%\"xGF(F)F)* &\"\"!F)F+F)F)\"\")!\"\"" }{TEXT -1 5 " when" }{XPPEDIT 18 0 "``(x^3-8 )" "6#-%!G6#,&*$%\"xG\"\"$\"\"\"\"\")!\"\"" }{TEXT -1 15 " is divided \+ by " }{XPPEDIT 18 0 "x^2" "6#*$%\"xG\"\"#" }{TEXT -1 7 ", while" } {XPPEDIT 18 0 "``(4*x-8)" "6#-%!G6#,&*&\"\"%\"\"\"%\"xGF)F)\"\")!\"\" " }{TEXT -1 22 " is the remainder when" }{XPPEDIT 18 0 "``(x^3-8)" "6# -%!G6#,&*$%\"xG\"\"$\"\"\"\"\")!\"\"" }{TEXT -1 15 " is divided by " } {XPPEDIT 18 0 "``(x^2+2*x)" "6#-%!G6#,&*$%\"xG\"\"#\"\"\"*&F)F*F(F*F* " }{TEXT -1 38 ". Finally, the 0 is the remainder when" }{XPPEDIT 18 0 "``(x^3-8)" "6#-%!G6#,&*$%\"xG\"\"$\"\"\"\"\")!\"\"" }{TEXT -1 14 " \+ is divided by" }{XPPEDIT 18 0 " ``(x^2+2*x+4)" "6#-%!G6#,(*$%\"xG\"\"# \"\"\"*&F)F*F(F*F*\"\"%F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "Thus we see that" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x^3-8)=(x-2)*(x^2+2*x+4)" "6#/,&*$%\"xG\"\"$\"\"\"\"\")!\"\"*&, &F&F(\"\"#F*F(,(*$F&F-F(*&F-F(F&F(F(\"\"%F(F(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "It is an \+ example of the following more general factorisation of a \"" }{TEXT 259 23 "difference of two cubes" }{TEXT -1 3 "\". " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "a^3-b^3=(a-b)*(a^2+a*b+b^2)" "6#/,&*$ %\"aG\"\"$\"\"\"*$%\"bGF'!\"\"*&,&F&F(F*F+F(,(*$F&\"\"#F(*&F&F(F*F(F(* $F*F0F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 71 "A companion formula to the previous formula is the factorisation of a \"" }{TEXT 259 16 "sum of two cubes" }{TEXT -1 2 "\"." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a^3+b^3=(a+b)*(a^2-a*b+b^2)" "6#/,&*$%\"aG\" \"$\"\"\"*$%\"bGF'F(*&,&F&F(F*F(F(,(*$F&\"\"#F(*&F&F(F*F(!\"\"*$F*F/F( F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 58 "These last two for mulas can be checked by multiplication. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Returning to the problem of the ev aluation of the " }{XPPEDIT 18 0 "Limit((x^3-8)/(x-2),x = 2)" "6#-%&Li mitG6$*&,&*$%\"xG\"\"$\"\"\"\"\")!\"\"F+,&F)F+\"\"#F-F-/F)F/" }{TEXT -1 18 ", we now see that " }{XPPEDIT 18 0 "(x^3-8)/(x-2)" "6#*&,&*$%\" xG\"\"$\"\"\"\"\")!\"\"F(,&F&F(\"\"#F*F*" }{TEXT -1 28 " can be writte n in the form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x^ 3-8)/(x-2)=(x-2)*(x^2+2*x+4)/(x-2)" "6#/*&,&*$%\"xG\"\"$\"\"\"\"\")!\" \"F),&F'F)\"\"#F+F+*(,&F'F)F-F+F),(*$F'F-F)*&F-F)F'F)F)\"\"%F)F),&F'F) F-F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 20 "For any real n umber " }{TEXT 336 1 "x" }{TEXT -1 41 " which is different from 2, the value of " }{XPPEDIT 18 0 "(x^3-8)/(x-2)" "6#*&,&*$%\"xG\"\"$\"\"\"\" \")!\"\"F(,&F&F(\"\"#F*F*" }{TEXT -1 42 " is the same as the value of \+ the quadratic" }{XPPEDIT 18 0 " ``(x^2+2*x+4)" "6#-%!G6#,(*$%\"xG\"\"# \"\"\"*&F)F*F(F*F*\"\"%F*" }{TEXT -1 13 ", because the" }{XPPEDIT 18 0 "``(x-2)" "6#-%!G6#,&%\"xG\"\"\"\"\"#!\"\"" }{TEXT -1 46 "'s at the \+ top and bottom cancel out, that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x^3-8)/(x-2) = (x^2+2*x+4)" "6#/*&,&*$%\"xG\"\"$\" \"\"\"\")!\"\"F),&F'F)\"\"#F+F+,(*$F'F-F)*&F-F)F'F)F)\"\"%F)" }{TEXT -1 7 ", when " }{XPPEDIT 18 0 "x<>2" "6#0%\"xG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "It is " }{TEXT 259 14 "very important" } {TEXT -1 30 " to include the qualification " }{XPPEDIT 18 0 "x<>2" "6# 0%\"xG\"\"#" }{TEXT -1 81 " along with this equation, because this lie s at the heart of the matter in hand. " }}{PARA 0 "" 0 "" {TEXT -1 23 "If we wish to evaluate " }{XPPEDIT 18 0 "f(x)=(x^3-8)/(x-2)" "6#/-%\" fG6#%\"xG*&,&*$F'\"\"$\"\"\"\"\")!\"\"F,,&F'F,\"\"#F.F." }{TEXT -1 18 " for any value of " }{TEXT 299 1 "x" }{TEXT -1 58 " other than 2, we \+ may as well use the quadratic expression" }{XPPEDIT 18 0 "``(x^2+2*x+4 )" "6#-%!G6#,(*$%\"xG\"\"#\"\"\"*&F)F*F(F*F*\"\"%F*" }{TEXT -1 10 " in stead. " }}{PARA 0 "" 0 "" {TEXT -1 34 "This applies, in particular, w hen " }{TEXT 300 1 "x" }{TEXT -1 16 " is close to 2. " }}{PARA 0 "" 0 "" {TEXT -1 18 "Now the graph of " }{XPPEDIT 18 0 "y=x^2+2*x+4" "6#/% \"yG,(*$%\"xG\"\"#\"\"\"*&F(F)F'F)F)\"\"%F)" }{TEXT -1 115 " is a comp lete parabola with no missing point in its graph. This explains the sh ape of the curve drawn previously. " }}{PARA 0 "" 0 "" {TEXT -1 6 "Sin ce " }{XPPEDIT 18 0 " x^2+2*x+4=(x+1)^2+3" "6#/,(*$%\"xG\"\"#\"\"\"*&F 'F(F&F(F(\"\"%F(,&*$,&F&F(F(F(F'F(\"\"$F(" }{TEXT -1 41 ", the vertex \+ of the parabola is the point" }{XPPEDIT 18 0 "``(-1,3)" "6#-%!G6$,$\" \"\"!\"\"\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "g := x -> x^2+2*x+4: 'g(x)'= g(x); \nplot(g(x),x=-4..3,y=-2..g(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,(*$)F'\"\"#\"\"\"F,*&F+F,F'F,F,\"\"%F," }}{PARA 13 "" 1 "" {GLPLOT2D 292 375 375 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!\"% \"\"!$\"#7F*7$$!3]LL$3#*>u%Q!#<$\"3\"3&eb?+y56!#;7$$!3Ym;z43m9PF0$\"3y u=AJ$Qp.\"F37$$!3;LLe/$f`c$F0$\"3g\"G7;Oo5e*F07$$!3PLL3K\"o]T$F0$\"3\\ \"[/E3aD$))F07$$!3]m;Hn7\\lKF0$\"3Q-[<#o]C8)F07$$!3WL$ekO9o7$F0$\"3(Hc t$\\$RL_(F07$$!3q**\\7oCA$)HF0$\"3'e\"e2e8Z$GF0$\"3S& H?Cs&>mjF07$$!3A+]iDGp'o#F0$\"3xZ[,)oK\\%eF07$$!3Cmm;u\"HW`#F0$\"3E(zL 0*GZa`F07$$!3?LL3n_J+CF0$\"3SmmAZG)3'\\F07$$!3q****\\sZL\\AF0$\"3+Ewxt t$3c%F07$$!3#*****\\PVt(4#F0$\"3zjlsv1-0UF07$$!3')****\\F#R;&>F0$\"3i' z;$>sh0RF07$$!3DL$e9(3(*==F0$\"3w@sH)G82n$F07$$!3[mm;k`@h;F0$\"3Wd+\"y v0sV$F07$$!3MmmmcddF:F0$\"3q0C-zhLyKF07$$!3!***\\7B67s8F0$\"3,6eFITZQJ F07$$!3smmmJu^M7F0$\"3#)HbvD%)*\\0$F07$$!3i**\\7tVa$3\"F0$\"37\\3Gi'zp +$F07$$!3_%**\\P>ByR*!#=$\"3pa)f!phi.IF07$$!3-jm;zp\"y*yF`r$\"3$)*zIXt \">WIF07$$!3ojm\"H-V._'F`r$\"3[Z;yE,3@JF07$$!3)RLL3F^X.&F`r$\"3iG)>3\" obYKF07$$!3SKLe97B\"\\$F`r$\"3*Q9?5rSOU$F07$$!31)**\\(olwZ@F`r$\"3*=S) e)pvlh$F07$$!3q.LLe%zy'p!#>$\"35?zCXv\\lQF07$$\"3g(******\\\\@-)F_t$\" 3wM+Ey%yo;%F07$$\"3%Q++v$opoAF`r$\"3yD[!4A4_]%F07$$\"3g0+voMf(o$F`r$\" 3Yywl\\@]t[F07$$\"3A)***\\ii.j_F`r$\"3G**Q]fFgH`F07$$\"3UGLL$oT'ymF`r$ \"3)*R-,%)3x\"y&F07$$\"3i3++DE5!>)F`r$\"31A.3N$)z3jF07$$\"3[jmT&)3rf&* F`r$\"3gr3@**G#e#oF07$$\"3*4++vW0d5\"F0$\"3:z6jJa*RV(F07$$\"3;L$3-\"Qf Y7F0$\"3-&Rm![P=Z!)F07$$\"3C+]PWF'QR\"F0$\"3p2m!*Q)y0t)F07$$\"3[LL$e/X y`\"F0$\"3C%zgmZd1W*F07$$\"3m**\\(=<\"e)o\"F0$\"3jugxro%G-\"F37$$\"3%y mmm(zvL=F0$\"3x'>KqU=I5\"F37$$\"3-nm\"zAAA)>F0$\"3%\\-o;%\\O*=\"F37$$ \"3LM$3-7d%H@F0$\"39kAs'=]$z7F37$$\"3#4++]p]ZE#F0$\"3%*)H]+rfeO\"F37$$ \"3$QL$e*R7)>CF0$\"3-#RM[o6&p9F37$$\"3'pmmmV,&eDF0$\"3MteZZKHm:F37$$\" 3<+](o(GP1FF0$\"3WmMC!**>Pn\"F37$$\"3g+]78Z!z%GF0$\"3If*G\"oqj!y\"F37$ $\"\"$F*$\"#>F*-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*Fb[l-%+AXESLABELSG6$Q \"x6\"Q\"yFg[l-%%VIEWG6$;F(Fgz;$!\"#F*Fiz" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 23 "In particular the point" }{XPPEDIT 18 0 " ``(2,12)" "6# -%!G6$\"\"#\"#7" }{TEXT -1 29 " lies on this parabola since " } {XPPEDIT 18 0 "x^2+2*x+4" "6#,(*$%\"xG\"\"#\"\"\"*&F&F'F%F'F'\"\"%F'" }{TEXT -1 23 " has the value 12 when " }{XPPEDIT 18 0 "x=2" "6#/%\"xG \"\"#" }{TEXT -1 24 ". Thus it is clear that " }{XPPEDIT 18 0 "Limit(` `(x^2+2*x+4),x = 2) = 12;" "6#/-%&LimitG6$-%!G6#,(*$%\"xG\"\"#\"\"\"*& F-F.F,F.F.\"\"%F./F,F-\"#7" }{TEXT -1 37 " by mere evaluation of the e xpression" }{XPPEDIT 18 0 "``(x^2+2*x+4)" "6#-%!G6#,(*$%\"xG\"\"#\"\" \"*&F)F*F(F*F*\"\"%F*" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "x=2" "6#/% \"xG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 19 "Since the va lues of" }{XPPEDIT 18 0 "``(x^2+2*x+4)" "6#-%!G6#,(*$%\"xG\"\"#\"\"\"* &F)F*F(F*F*\"\"%F*" }{TEXT -1 35 " get progressively closer to 12 as \+ " }{TEXT 301 1 "x" }{TEXT -1 45 " approaches 2, the same conclusion ho lds for " }{XPPEDIT 18 0 "f(x) = (x^3-8)/(x-2)" "6#/-%\"fG6#%\"xG*&,&* $F'\"\"$\"\"\"\"\")!\"\"F,,&F'F,\"\"#F.F." }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 29 "In conclusion, we may write: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit((x^3-8)/(x-2),x = 2) = Limit((x -2)*(x^2+2*x+4)/(x-2),x = 2);" "6#/-%&LimitG6$*&,&*$%\"xG\"\"$\"\"\"\" \")!\"\"F,,&F*F,\"\"#F.F./F*F0-F%6$*(,&F*F,F0F.F,,(*$F*F0F,*&F0F,F*F,F ,\"\"%F,F,,&F*F,F0F.F./F*F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Limit(``(x^2+2*x+4),x = 2)" "6#/%!G- %&LimitG6$-F$6#,(*$%\"xG\"\"#\"\"\"*&F-F.F,F.F.\"\"%F./F,F-" } {XPPEDIT 18 0 "``=12" "6#/%!G\"#7" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "factor(x^3-8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# *&,&%\"xG\"\"\"\"\"#!\"\"F&,(*$)F%F'F&F&*&F'F&F%F&F&\"\"%F&F&" }}} {PARA 0 "" 0 "" {TEXT -1 30 "Maple ignores the restriction " } {XPPEDIT 18 0 "x <> 2;" "6#0%\"xG\"\"#" }{TEXT -1 27 ", if we ask it t o simplify " }{XPPEDIT 18 0 "(x^3-8)/(x-2);" "6#*&,&*$%\"xG\"\"$\"\"\" \"\")!\"\"F(,&F&F(\"\"#F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "simplify((x^3-8)/( x-2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"#\"\"\"F(*&F'F (F&F(F(\"\"%F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Maple can evaluate limits using the " }{TEXT 0 5 "Limit" }{TEXT -1 4 " or " }{TEXT 0 5 "limit" }{TEXT -1 10 " command. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "(x^3-8)/(x-2);\nlimit(%,x=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#* &,&*$)%\"xG\"\"$\"\"\"F)\"\")!\"\"F),&F'F)\"\"#F+F+" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Using " }{TEXT 0 5 "Limit" }{TEXT -1 9 " with an " } {TEXT 259 9 "uppercase" }{TEXT -1 1 " " }{TEXT 303 1 "L" }{TEXT -1 29 " just prints the limit in an " }{TEXT 259 11 "unevaluated" }{TEXT -1 7 " form. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "The command " }{TEXT 0 5 "value" }{TEXT -1 51 " can then be use d to obtain the value of the limit." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Limit((x^3-8)/(x-2),x=2);\n value(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&,&*$)%\"xG \"\"$\"\"\"F,\"\")!\"\"F,,&F*F,\"\"#F.F./F*F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "Examples of finding limits algebraically " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 306 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 18 "Find the value of " }{XPPEDIT 18 0 "Limit((x^2+2*x-3)/(x-1),x = 1);" "6#-%&LimitG6$*&,(*$%\"xG\"\"#\"\"\"*&F*F+F)F+F+\"\"$!\"\"F+,&F)F+F+F. F./F)F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 307 8 "Solution" } {TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Lim it((x^2+2*x-3)/(x-1),x = 1) = Limit((x-1)*(x+3)/(x-1),x = 1);" "6#/-%& LimitG6$*&,(*$%\"xG\"\"#\"\"\"*&F+F,F*F,F,\"\"$!\"\"F,,&F*F,F,F/F//F*F ,-F%6$*(,&F*F,F,F/F,,&F*F,F.F,F,,&F*F,F,F/F//F*F," }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Limit(``(x+3),x \+ = 1);" "6#/%!G-%&LimitG6$-F$6#,&%\"xG\"\"\"\"\"$F,/F+F," }{XPPEDIT 18 0 "``=4" "6#/%!G\"\"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 305 11 "Explanation" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 18 "The f actorisation " }{XPPEDIT 18 0 "x^2+2*x-3=(x-1)*(x+3)" "6#/,(*$%\"xG\" \"#\"\"\"*&F'F(F&F(F(\"\"$!\"\"*&,&F&F(F(F+F(,&F&F(F*F(F(" }{TEXT -1 14 ", shows that " }{XPPEDIT 18 0 "(x^2+2*x-3)/(x-1))=x+3" "6#/*&,(*$ %\"xG\"\"#\"\"\"*&F(F)F'F)F)\"\"$!\"\"F),&F'F)F)F,F,,&F'F)F+F)" } {TEXT -1 7 ", when " }{XPPEDIT 18 0 "x<>1" "6#0%\"xG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "Since the graph of " }{XPPEDIT 18 0 "y=x+3" "6#/%\"yG,&%\"xG\"\"\"\"\"$F'" }{TEXT -1 64 " has no brea ks (It is an infinite straight line), the values of " }{XPPEDIT 18 0 " x+3" "6#,&%\"xG\"\"\"\"\"$F%" }{TEXT -1 15 " approach 4 as " }{TEXT 304 1 "x" }{TEXT -1 51 " approaches 1. Hence the same conclusion holds for " }{XPPEDIT 18 0 "(x^2+2*x-3)/(x-1)" "6#*&,(*$%\"xG\"\"#\"\"\"*&F 'F(F&F(F(\"\"$!\"\"F(,&F&F(F(F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 "y=(x^2+2*x-3)/(x-1)" "6#/% \"yG*&,(*$%\"xG\"\"#\"\"\"*&F)F*F(F*F*\"\"$!\"\"F*,&F(F*F*F-F-" } {TEXT -1 51 " is a straight line with a missing point or hole at" } {XPPEDIT 18 0 "``(1,4)" "6#-%!G6$\"\"\"\"\"%" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "f := x -> (x^2+2*x-3)/(x-1):\n'f(x)'=f(x);\nplot([f(x),[[1,4]]$2] ,x=-2..4,y=0..f(4),style=[line,point$2],\n symbol=circle,symbols ize=[15,18],color=[red,brown$2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"fG6#%\"xG*&,(*$)F'\"\"#\"\"\"F-*&F,F-F'F-F-\"\"$!\"\"F-,&F'F-F-F0F 0" }}{PARA 13 "" 1 "" {GLPLOT2D 338 342 342 {PLOTDATA 2 "6(-%'CURVESG6 &7S7$$!\"#\"\"!$\"\"\"F*7$$!3!******\\2<#p=!#<$\"36+++DHyI6F07$$!31++D ^NUbF07$$!3M++]i83V()!#=$\"3>++vj=pD@F07 $$!3B******\\V'zV(FY$\"3')*****\\c.iD#F07$$!3%)*****\\d;%)G'FY$\"3C++] U$e6P#F07$$!3#*)*****\\!)H%*\\FY$\"3))*****\\>q0]#F07$$!3Q+++]d'[p$FY$ \"3=+++DM^IEF07$$!3/******\\>iUCFY$\"3))*****\\!ytbFF07$$!3B++]7YY08FY $\"3?++vQNXpGF07$$\"3%z-+++XDn%!#?$\"3.+++XDn/IF07$$\"3C++++y?#>\"FY$ \"3e******z2A>JF07$$\"3h****\\(3wY_#FY$\"3'****\\(3wY_KF07$$\"3F)***** *HOTq$FY$\"3Q******HOTqLF07$$\"3I,+](3\">)*\\FY$\"37++v3\">)*\\$F07$$ \"3_,+]isVIiFY$\"3:++DEP/BOF07$$\"3&=++](o:;vFY$\"3k++](o:;v$F07$$\"3# >++v$)[op)FY$\"3u***\\P)[opQF07$$\"3W*****\\i%Qq**FY$\"3))z+]i%Qq*RF07 $$\"3&****\\(QIKH6F0$\"3%3+](QIKHTF07$$\"3#****\\7:xWC\"F0$\"3.***\\7: xWC%F07$$\"37++]Zn%)o8F0$\"3M)***\\Zn%)oVF07$$\"3y******4FL(\\\"F0$\"3 M******4FL(\\%F07$$\"3#)****\\d6.B;F0$\"3r++]d6.BYF07$$\"3(****\\(o3lW ***\\P#4 JBcF07$$\"3u*****\\KCnu#F0$\"3u*****\\KCnu&F07$$\"3s***\\(=n#f(GF0$\"3 s***\\(=n#f(eF07$$\"3P+++!)RO+IF0$\"3P+++!)RO+gF07$$\"30++]_!>w7$F0$\" 30++]_!>w7'F07$$\"3O++v)Q?QD$F0$\"3C,+v)Q?QD'F07$$\"3G+++5jypLF0$\"3;, ++5jypjF07$$\"3<++]Ujp-NF0$\"31,+]Ujp-lF07$$\"3++++gEd@OF0$\"3++++gEd@ mF07$$\"39++v3'>$[PF0$\"3E***\\(3'>$[nF07$$\"37++D6EjpQF0$\"37++D6Ejpo F07$$\"\"%F*$\"\"(F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fa[l-%&STYLEG 6#%%LINEG-%'SYMBOLG6$%(DEFAULTG\"#:-F$6&7#7$F+Ffz-F[[l6&F][l$\")#)eqkF `[l$\"))eqk\"F`[lFc\\l-Fc[l6#%&POINTG-Fg[l6$Fi[l\"#=-F$6&F]\\lF_\\lFe \\lFf[l-%+AXESLABELSG6$Q\"x6\"Q\"yFa]l-Fg[l6#%'CIRCLEG-%%VIEWG6$;F(Ffz ;Fa[lFhz" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur ve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Limit((x^2+2*x-3)/(x-1),x=1);\n``=s implify(%);\n``=value(rhs(%));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% &LimitG6$*&,(*$)%\"xG\"\"#\"\"\"F,*&F+F,F*F,F,\"\"$!\"\"F,,&F*F,F,F/F/ /F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,&%\"xG\"\"\" \"\"$F*/F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 312 8 "Question" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 18 "Find the value of " } {XPPEDIT 18 0 "Limit((4*x^2-1)/(2*x-1),x = 1/2);" "6#-%&LimitG6$*&,&*& \"\"%\"\"\"*$%\"xG\"\"#F*F*F*!\"\"F*,&*&F-F*F,F*F*F*F.F./F,*&F*F*F-F. " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 313 8 "Solution" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit((4*x^2 -1)/(2*x-1),x = 1/2)=Limit((2*x-1)*(2*x+1)/(2*x-1),x = 1/2)" "6#/-%&Li mitG6$*&,&*&\"\"%\"\"\"*$%\"xG\"\"#F+F+F+!\"\"F+,&*&F.F+F-F+F+F+F/F//F -*&F+F+F.F/-F%6$*(,&*&F.F+F-F+F+F+F/F+,&*&F.F+F-F+F+F+F+F+,&*&F.F+F-F+ F+F+F/F//F-*&F+F+F.F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Limit(``(2*x +1),x = 1/2);" "6#/%!G-%&LimitG6$-F$6#,&*&\"\"#\"\"\"%\"xGF-F-F-F-/F.* &F-F-F,!\"\"" }{XPPEDIT 18 0 "`` = 2;" "6#/%!G\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 311 11 "Explanation" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 19 "The factorisation " }{XPPEDIT 18 0 "4*x^2-1=(2 *x-1)*(2*x+1)" "6#/,&*&\"\"%\"\"\"*$%\"xG\"\"#F'F'F'!\"\"*&,&*&F*F'F)F 'F'F'F+F',&*&F*F'F)F'F'F'F'F'" }{TEXT -1 14 ", shows that " } {XPPEDIT 18 0 "(4*x^2-1)/(2*x-1)=2*x+1" "6#/*&,&*&\"\"%\"\"\"*$%\"xG\" \"#F(F(F(!\"\"F(,&*&F+F(F*F(F(F(F,F,,&*&F+F(F*F(F(F(F(" }{TEXT -1 7 ", when " }{XPPEDIT 18 0 "x<>1/2" "6#0%\"xG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "Since the graph of " } {XPPEDIT 18 0 "y = 2*x+1;" "6#/%\"yG,&*&\"\"#\"\"\"%\"xGF(F(F(F(" } {TEXT -1 65 " has no breaks (It is an infinite straight line), the val ues of " }{XPPEDIT 18 0 "2*x+1;" "6#,&*&\"\"#\"\"\"%\"xGF&F&F&F&" } {TEXT -1 15 " approach 2 as " }{TEXT 310 1 "x" }{TEXT -1 12 " approach es " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 38 ". Hen ce the same conclusion holds for " }{XPPEDIT 18 0 "(4*x^2-1)/(2*x-1)" "6#*&,&*&\"\"%\"\"\"*$%\"xG\"\"#F'F'F'!\"\"F',&*&F*F'F)F'F'F'F+F+" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "The graph of " } {XPPEDIT 18 0 "y=(4*x^2-1)/(2*x-1)" "6#/%\"yG*&,&*&\"\"%\"\"\"*$%\"xG \"\"#F)F)F)!\"\"F),&*&F,F)F+F)F)F)F-F-" }{TEXT -1 51 " is a straight l ine with a missing point or hole at" }{XPPEDIT 18 0 "``(1/2, 2);" "6#- %!G6$*&\"\"\"F'\"\"#!\"\"F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "f := x -> (4*x^2- 1)/(2*x-1):\n'f(x)'=f(x);\nplot([f(x),[[1/2,2]]$2],x=-1..3,style=[line ,point$2],\n symbol=circle,symbolsize=[15,18],color=[red,brown$2 ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&,&*&\"\"%\"\"\" )F'\"\"#F,F,F,!\"\"F,,&*&F.F,F'F,F,F,F/F/" }}{PARA 13 "" 1 "" {GLPLOT2D 311 360 360 {PLOTDATA 2 "6(-%'CURVESG6&7S7$$!\"\"\"\"!F(7$$! 3PLLLLQ6G\"*!#=$!3ummmmwAc#)F.7$$!3immmT.\\p$)F.$!3LMLL$o!)*QnF.7$$!3L LLL$))Qj^(F.$!3xnmmmxnK]F.7$$!3ULLL$=Kvl'F.$!3#ommmOk]J$F.7$$!3hnmmTs! G!eF.$!3&\\LLL[9cg\"F.7$$!3iLLL3yO5]F.$!3U.nmm;ct?!#?7$$!3i+++vE%)*=%F .$\"3v)*****\\YJ?;F.7$$!3)RLL$3WDTLF.$\"3/KLL$=\"\\$\"3aLLLL76#G)F.7$$\"3HI#*******H,Q!#@$\"3&)******f-w+5!#<7$$\"3 Q(*******\\*3q)F\\o$\"3%*********y,u6Feo7$$\"3!********p=\\q\"F.$\"3)* ******RP)4M\"Feo7$$\"3_mmm\"fBIY#F.$\"3ILLL=Zg#\\\"Feo7$$\"3yKLLLO[kLF .$\"3cmmmEn*Gn\"Feo7$$\"3.KLLL&Q\"GTF.$\"3=mmm1xiD=Feo7$$\"3+*****\\s] k,&F.$\"3M>++X,H.?Feo7$$\"3WJLLLvv-eF.$\"3Immm1:bg@Feo7$$\"3'3++]sgam' F.$\"3<+++X@4LBFeo7$$\"3G+++v\"ep[(F.$\"3]+++N;R(\\#Feo7$$\"3#QLLLe/TM )F.$\"3wmmm;4#)oEFeo7$$\"39LLLeDBJ\"*F.$\"3jmmm6lCEGFeo7$$\"3Immm;kD!) **F.$\"3ELLL$G^g*HFeo7$$\"3Mmm;f`@'3\"Feo$\"3oKLL=2VsJFeo7$$\"3y****\\ nZ)H;\"Feo$\"3f*****\\`pfK$Feo7$$\"3YmmmJy*eC\"Feo$\"3!HLLLm&z\"\\$Feo 7$$\"3')******R^bJ8Feo$\"3s******z-6jOFeo7$$\"3f*****\\5a`T\"Feo$\"3<* *****4#32$QFeo7$$\"3o****\\7RV'\\\"Feo$\"3#*)****\\#y'G*RFeo7$$\"3k*** **\\@fke\"Feo$\"3G******H%=H<%Feo7$$\"3/LLL`4Nn;Feo$\"35mmm1>qMVFeo7$$ \"3#*******\\,s`$=Feo$\"3'HLL$e p'Rm%Feo7$$\"3$*******pfa<>Feo$\"3')******R>4N[Feo7$$\"3#HLLeg`!)*>Feo $\"3#emm;@2h*\\Feo7$$\"3w****\\#G2A3#Feo$\"3S+++lXTk^Feo7$$\"3;LLL$)G[ k@Feo$\"3Lmmmmd'*G`Feo7$$\"3#)****\\7yh]AFeo$\"3j*****\\iN7]&Feo7$$\"3 xmmm')fdLBFeo$\"3aLLLt>:ncFeo7$$\"3bmmm,FT=CFeo$\"35LLL.a#o$eFeo7$$\"3 FLL$e#pa-DFeo$\"3ammm^Q40gFeo7$$\"3!*******Rv&)zDFeo$\"3y******z]rfhFe o7$$\"3ILLLGUYoEFeo$\"3gmmmc%GpL'Feo7$$\"3_mmm1^rZFFeo$\"3/LLL8-V&\\'F eo7$$\"34++]sI@KGFeo$\"3=+++XhUkmFeo7$$\"34++]2%)38HFeo$\"3H*****\\\"o " 0 "" {MPLTEXT 1 0 66 "Limit((4*x^2-1)/(2 *x-1),x=1/2);\n``=simplify(%);\n``=value(rhs(%));\n" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%&LimitG6$*&,&*&\"\"%\"\"\")%\"xG\"\"#F*F*F*!\"\"F*, &*&F-F*F,F*F*F*F.F./F,#F*F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%& LimitG6$,&*&\"\"#\"\"\"%\"xGF+F+F+F+/F,#F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT 315 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 " " 0 "" {TEXT -1 18 "Find the value of " }{XPPEDIT 18 0 "Limit((x^2+2*x )/(x^2+x-2),x = -2);" "6#-%&LimitG6$*&,&*$%\"xG\"\"#\"\"\"*&F*F+F)F+F+ F+,(*$F)F*F+F)F+F*!\"\"F//F),$F*F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT 316 8 "Solution" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit((x^2+2*x)/(x^2+x-2),x = -2) = Limit(x*( x+2)/((x-1)*(x+2)),x = -2);" "6#/-%&LimitG6$*&,&*$%\"xG\"\"#\"\"\"*&F+ F,F*F,F,F,,(*$F*F+F,F*F,F+!\"\"F0/F*,$F+F0-F%6$*(F*F,,&F*F,F+F,F,*&,&F *F,F,F0F,,&F*F,F+F,F,F0/F*,$F+F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Limit(x/(x-1),x = -2);" "6#/%!G-%& LimitG6$*&%\"xG\"\"\",&F)F*F*!\"\"F,/F),$\"\"#F," }{XPPEDIT 18 0 "``=2 /3" "6#/%!G*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT 314 11 "Explanation" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x^2+2*x)/(x^2+x-2)=x*(x+2)/((x-1)*(x+2 ))" "6#/*&,&*$%\"xG\"\"#\"\"\"*&F(F)F'F)F)F),(*$F'F(F)F'F)F(!\"\"F-*(F 'F),&F'F)F(F)F)*&,&F'F)F)F-F),&F'F)F(F)F)F-" }{XPPEDIT 18 0 "``=x/(x-1 )" "6#/%!G*&%\"xG\"\"\",&F&F'F'!\"\"F)" }{TEXT -1 7 ", when " } {XPPEDIT 18 0 "x <> -2;" "6#0%\"xG,$\"\"#!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 20 "Since the graph of " }{XPPEDIT 18 0 "y = x/(x-1);" "6#/%\"yG*&%\"xG\"\"\",&F&F'F'!\"\"F)" }{TEXT -1 20 " has n o breaks near " }{XPPEDIT 18 0 "x=-2" "6#/%\"xG,$\"\"#!\"\"" }{TEXT -1 16 ", the values of " }{XPPEDIT 18 0 "x/(x-1)" "6#*&%\"xG\"\"\",&F$ F%F%!\"\"F'" }{TEXT -1 10 " approach " }{XPPEDIT 18 0 "(-2)/(-3) = 2/3 ;" "6#/*&,$\"\"#!\"\"\"\"\",$\"\"$F'F'*&F&F(F*F'" }{TEXT -1 4 " as " } {TEXT 317 1 "x" }{TEXT -1 16 " approaches -2. " }}{PARA 0 "" 0 "" {TEXT -1 20 "Hence the values of " }{XPPEDIT 18 0 "x*(x+2)/((x-1)*(x+2 ))" "6#*(%\"xG\"\"\",&F$F%\"\"#F%F%*&,&F$F%F%!\"\"F%,&F$F%F'F%F%F*" } {TEXT -1 15 " also approach " }{XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\" \"$!\"\"" }{TEXT -1 4 " as " }{TEXT 318 1 "x" }{TEXT -1 16 " approache s -2. " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 15 ": The graph of " }{XPPEDIT 18 0 "y = x/(x-1)" "6#/%\"yG*&%\"xG\"\"\",&F&F'F'!\"\" F)" }{TEXT -1 54 " is a rectangular hyperbola with a vertical asymptot e " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 11 ", so there " } {TEXT 259 2 "is" }{TEXT -1 23 " a break in graph when " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 39 ", but this is well away from th e value " }{XPPEDIT 18 0 "x=-2" "6#/%\"xG,$\"\"#!\"\"" }{TEXT -1 34 " \+ at which the limit is evaluated. " }}{PARA 0 "" 0 "" {TEXT -1 13 "The \+ graph of " }{XPPEDIT 18 0 "y=x*(x+2)/((x-1)*(x+2))" "6#/%\"yG*(%\"xG\" \"\",&F&F'\"\"#F'F'*&,&F&F'F'!\"\"F',&F&F'F)F'F'F," }{TEXT -1 49 " can be formed by \"punching a hole\" in the graph " }{XPPEDIT 18 0 "y=x/( x-1)" "6#/%\"yG*&%\"xG\"\"\",&F&F'F'!\"\"F)" }{TEXT -1 14 " at the poi nt " }{XPPEDIT 18 0 "``(-2,2/3);" "6#-%!G6$,$\"\"#!\"\"*&F'\"\"\"\"\"$ F(" }{TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 299 "f := x -> x/(x-1):\n'f(x)'=f(x);\np1 := \+ plot(f(x),x=-3..3,y=-2..4,discont=true,color=red):\np2 := plot([[[-2,2 /3]]$2],style=[point$2],\n symbol=[circle$2],symbolsize=[15,18], color=[brown$2]):\np3 := plots[implicitplot](\{x=1,y=1\},x=-3..3,y=-2. .4,color=black,linestyle=3):\nplots[display]([p1,p2,p3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&F'\"\"\",&F'F)F)!\"\"F+" }} {PARA 13 "" 1 "" {GLPLOT2D 348 295 295 {PLOTDATA 2 "6)-%'CURVESG6%7gn7 $$!\"$\"\"!$\"3++++++++v!#=7$$!31;TY$Q6G\"H!#<$\"3B&\\$\\dGHWuF-7$$!3 \"HC6W.\\p$GF1$\"3!p*[k>Ew$R(F-7$$!3m#)eq))Qj^FF1$\"3mxIt.Z\\MtF-7$$!3 s.Z$)=KvlEF1$\"3)3H=%zw/ssF-7$$!3QXizC2G!e#F1$\"3=Wz/HH#p?(F-7$$!3+yZ:vqF-7$$!3KX@$=WDT L#F1$\"3$=\\\"GGFr+qF-7$$!3'=pD'e(Q&\\AF1$\"3(4PqG=SE#pF-7$$!3?qG#z&4` i@F1$\"3nT')H' **>F1$\"3mfG9[UCmmF-7$$!3`80j^5*H\">F1$\"3=)palR-rc'F-7$$!3zPdvJ\"3&H= F1$\"3OEReOo\"eY'F-7$$!3%oy-Fk(p`&ojF-7$$!39R8nQ;bj;F1$ \"3'f;q)oVhXiF-7$$!3Q()ey[h=(e\"F1$\"3o<&p+(ozMhF-7$$!3knCvH\\N)\\\"F1 $\"3at&fW:mt*fF-7$$!3]!3P!\\Us>9F1$\"3aWzL(o(HneF-7$$!3:>)***HRXL8F1$ \"3,F$>p?2Xr&F-7$$!3wVI7&=/8D\"F1$\"3af.KJ-8ebF-7$$!3A#G=Wa*el6F1$\"3I &[+=y>BQ&F-7$$!3S^j.Zn(o3\"F1$\"364+1j,:3_F-7$$!3xr.LhV(>+\"F1$\"3(=-. b.J\\+&F-7$$!3SiliRk%y8*F-$\"3q\\rvx;vuZF-7$$!3QtZWdB:q$)F-$\"3e)))o+, (QcXF-7$$!3!3!=-<<-TvF-$\"3=GWo1y2*H%F-7$$!3'yKt\\j[Wo'F-$\"38)=%)fo%R 1SF-7$$!3!=JIi)*ek%eF-$\"35T)fq!>W*o$F-7$$!3E6lW74mN]F-$\"3Ed<5P]9\\LF -7$$!3D*)oz))ySNTF-$\"32Oxj>ncDHF-7$$!3pNpn10\\ELF-$\"3'fG#fL*[h\\#F-7 $$!3+/eIT&)ziCF-$\"3m!Qfn**>h(>F-7$$!3?)383Dl,o\"F-$\"3e*o*R%Hx%Q9F-7$ $!3)H*=jPMSX#)!#>$\"3&>ExlJCth(Ffv7$$!3wsvuj')RY>!#?$\"365L`bvhU>F\\w7 $$\"3K8*owyF2A)Ffv$!3)=(4N/O1d*)Ffv7$$\"3;)3meyG[k\"F-$!3gDpPVcjo>F-7$ $\"3TI2Cw!yh]#F-$!3,2/#[jDVM$F-7$$\"3!4.jm\")fdL$F-$!3V_]Nv:Y0]F-7$$\" 3!yZ!Rlp7%=%F-$!3K18r)3BV>(F-7$$\"3BK^z0#pa-&F-$!3N\")*[C%)R-,\"F17$$ \"3WR@IY`d)z&F-$!3e!H'*z-X,Q\"F17$$\"3AsjIGAk%o'F-$!3o.v6@dE;?F17$$\"3 yR4X55:xuF-$!3)ehB%>FxjHF17$$\"3(\\!o^n18A$)F-$!3IQr`s'R*f\\F17$$\"3.R -\">M2ls)F-$!3#>.tQh?C&oF17$$\"33tOI;S)38*F-$!3%oU\\'=Wf]5!#;7$$\"3oax A(*H;[$*F-$!3&eVHi&e7M9F[[l7$$\"3GO=:y>Wl&*F-$!3JhZVy')=,AF[[l7$$\"3kx Qho93u'*F-$!3EVQ.I9DoHF[[l7$$\"3)y\"f2f4s#y*F-$!3g%G89swB]%F[[l7$$\"3b QpI/2/P)*F-$!38kyOI<]OgF[[l7$$\"37ez`\\/O\"*)*F-$!3@BE+,4v/\"*F[[l7$$ \"3\"zY`@K?&=**F-$!3kwQ>%*)*H<7!#:7$$\"3!)y*oZ>!oX**F-$!3r2=Zj\"\\4$=F _]l7$$\"3q*[%Qn+%G(**F-$!3Ogb.gU*=n$F_]l7$$\"2%******R********F1$!3KiT 7Ummm;!\"*7gn7$$\"30+++1+++5F1$\"3KiT7immm;F_^l7$$\"3vH88GBO,5F1$\"3i$ 4z>Zv.N(F_]l7$$\"3WfEE]Ys-5F1$\"3MDU<=e>!o$F_]l7$$\"39*)RRsp3/5F1$\"3u jC;3!*zcCF_]l7$$\"3%)=`_%H\\a+\"F1$\"3sc*R'H**4X=F_]l7$$\"3ByzyQR<35F1 $\"3x1d5-/SL7F_]l7$$\"3iP10$e)*3,\"F1$\"3;O-f*p/bF*F[[l7$$\"3TcfdryM;5 F1$\"37$>#fbU+v75grz@5F1$\"3'RxVEh`xo%F[[l7$$\"3w7>:PdpK5 F1$\"3Y?B1*o-&eJF[[l7$$\"3b]D?9VfV5F1$\"3'*pT.Ar(QR#F[[l7$$\"3Q?tV,*fD 1\"F1$\"3Qgn^\"*zY)p\"F[[l7$$\"3?!4s')[D:3\"F1$\"3![j\"\\\\-hE8F[[l7$$ \"3<%yg91$=C6F1$\"3SqZNL'RBr;\"F1$\"3cn*Rpm-O)pF17$$ \"3%y3(GV'f)47F1$\"3'o0JCe*3ldF17$$\"3i))[$[h\"[\\7F1$\"3N.2m'R6$3]F17 $$\"3Mw%y8(y]!H\"F1$\"385Cr?vCUWF17$$\"3a@Xe%GPHL\"F1$\"3@)o^)y'oN+%F1 7$$\"373V7E1Bv8F1$\"3e.:mmw-lOF17$$\"3;j/TEXt=9F1$\"3u(o!*z%z9)Q$F17$$ \"35v@Y&y_qX\"F1$\"3+)f/?4Jz=$F17$$\"3!*H%*\\p+>+:F1$\"3ouP!4,S#**HF17 $$\"3w'[p$zW]V:F1$\"3p!oG*Q7\"*RGF17$$\"3QiUCRfC&e\"F1$\"3gt!)fjLo3FF1 7$$\"3!*zQr$=^Ji\"F1$\"3?)*p&=,ZZg#F17$$\"39%*>m&=C#o;F1$\"39*3t/v.l\\ #F17$$\"33YuaIpS1ks=L #F17$$\"33`iH!)y8!z\"F1$\"3GWQv'*=glAF17$$\"32\"=](RIFL=F1$\"38??/Ho3+ AF17$$\"3Scp77zMu=F1$\"3qU)oY[4P9#F17$$\"3D^]TK_?<>F1$\"3#)*f0&H&o-4#F 17$$\"3+#)p/J;cc>F1$\"3'>7(RY4TX?F17$$\"3)\\HOQ#G,**>F1$\"3y`Mk^\"))4+ #F17$$\"3k5SX#o2J/#F1$\"3yP,\\%zt'e>F17$$\"3gAb]'Q#\\\"3#F1$\"3))>rj5# [Y#>F17$$\"3v'[k%=*[H7#F1$\"37hQecC^!*=F17$$\"3MnE]svxl@F1$\"3wi;Onlzd =F17$$\"33pp([0xw?#F1$\"33!yob$f.G=F17$$\"33\\`]ep@[AF1$\"3*p7^mxU6!=F 17$$\"3I6.i4'HKH#F1$\"3wMFg!*yDt%po2goP#F1$\"3,O*)39-HEW!o-*\\#F1$\"3C/I%)y% *4n;F17$$\"3J*oEEk.6a#F1$\"3cOAyCc))[;F17$$\"3GU*>HWTAe#F1$\"3%)\\%\\U z9?j\"F17$$\"3>tSP2*3`i#F1$\"3(3AW\\ln_h\"F17$$\"3apHL%*zymEF1$\"3e1&> _Kc**f\"F17$$\"3L9dq^j?4FF1$\"3)f+HR'o1&e\"F17$$\"3?YGmjMF^FF1$\"3[*4K *[I,r:F17$$\"3g8-jq(G**y#F1$\"3ONnX\\9oe:F17$$\"3SqRm9@BMGF1$\"3gt)o/F (=X:F17$$\"3)4w6PbdQ(GF1$\"3***)\\&R\\eO`\"F17$$\"3]!o,l`1h\"HF1$\"37 \\G+\\;*=_\"F17$$\"3wn.)Q?Wl&HF1$\"3]^#4EC06^\"F17$$\"\"$F*$\"3+++++++ +:F1-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*F_am-F$6&7#7$$!\"#F*$\"3Immmm mmmmF--Fi`m6&F[am$\")#)eqkF^am$\"))eqk\"F^amF\\bm-%&STYLEG6#%&POINTG-% 'SYMBOLG6$%'CIRCLEG\"#:-F$6&FbamFhamF^bm-Fcbm6$Febm\"#=-F$6V7$7$F($\" \"\"F*7$$!3))************zGF1F`cm7$7$$!3!)************fFF1F`cmFbcm7$Ff cm7$$!3o************REF1F`cm7$7$$!3d************>DF1F`cmFjcm7$F^dm7$$! 3Y*************R#F1F`cm7$7$$!3O************zAF1F`cmFbdm7$Ffdm7$$!3E*** *********f@F1F`cm7$7$$!3:************R?F1F`cmFjdm7$F^em7$$!3E********* ***>>F1F`cm7$7$$!3;*************z\"F1F`cmFbem7$Ffem7$$!3E************z ;F1F`cm7$7$$!3;************f:F1F`cmFjem7$F^fm7$$!3G************R9F1F`c m7$7$$!3<************>8F1F`cmFbfm7$Fffm7$$!3G*************>\"F1F`cm7$7 $$!3=************z5F1F`cmFjfm7$F^gm7$$!3'H************f*F-F`cm7$7$$!3# >************R)F-F`cmFbgm7$Ffgm7$$!33$************>(F-F`cm7$7$$!3+#*** **********fF-F`cmFjgm7$F^hm7$$!3;$************z%F-F`cm7$7$$!33#******* *****f$F-F`cmFbhm7$Ffhm7$$!3C$************R#F-F`cm7$7$$!3=#*********** *>\"F-F`cmFjhm7$F^im7$$\"3CR4vZ\"Q8m'!#LF`cm7$7$$\"3s2++++++7F-F`cmFbi m7$Fgim7$$\"3c1++++++CF-F`cm7$7$$\"3i2++++++OF-F`cmF[jm7$F_jm7$$\"3[1+ +++++[F-F`cm7$7$$\"3a2++++++gF-F`cmFcjm7$Fgjm7$$\"3S1++++++sF-F`cm7$7$ $\"3Y2++++++%)F-F`cmF[[n7$F_[n7$$\"3I1++++++'*F-F`cm7$7$$\"3u++++++!3 \"F1F`cmFc[n7$Fg[n7$$\"3g+++++++7F1F`cm7$7$$\"3t++++++?8F1F`cmF[\\n7$F _\\n7$$\"3h++++++S9F1F`cm7$7$$\"3s++++++g:F1F`cmFc\\n7$Fg\\n7$$\"3g+++ +++!o\"F1F`cm7$7$$\"3q+++++++=F1F`cmF[]n7$F_]n7$$\"3#3++++++#>F1F`cm7$ 7$$\"3#4++++++/#F1F`cmFc]n7$Fg]n7$$\"3/,+++++g@F1F`cm7$7$$\"39,+++++!G #F1F`cmF[^n7$F_^n7$$\"3C,++++++CF1F`cm7$7$$\"3N,+++++?DF1F`cmFc^n7$Fg^ n7$$\"3Y,+++++SEF1F`cm7$7$$\"3c,+++++gFF1F`cmF[_n7$F__n7$$\"3m,+++++!) GF1F`cm7$7$$\"3y,++++++IF1F`cmFc_n-Fi`m6&F[amF*F*F*-%*LINESTYLEG6#Fe`m -F$6V7$7$F`cmFdam7$F`cm$!3[************>>F17$7$F`cm$!3C++++++g6F1F]fn7$F_fn7$F`cmF\\\\n7$7$F`cm$\"3))**** ********f8F1Fcfn7$Fefn7$F`cmFd\\n7$7$F`cm$\"33+++++++;F1Fifn7$7$F`cm$ \"3')*************f\"F17$F`cmF\\]n7$7$F`cm$\"3&)************R=F1Fbgn7$ Fdgn7$F`cmFd]n7$7$F`cm$\"33++++++!3#F1Fhgn7$Fjgn7$F`cmF\\^n7$7$F`cm$\" 3G++++++?BF1F^hn7$F`hn7$F`cmFd^n7$7$F`cm$\"3]++++++gDF1Fdhn7$Ffhn7$F`c mF\\_n7$7$F`cm$\"3q+++++++GF1Fjhn7$F\\in7$F`cmFd_n7$7$F`cm$\"3#4++++++ /$F1F`in7$Fbin7$F`cm$\"3)=++++++7$F17$7$F`cm$\"3e,+++++!G$F1Ffin7$7$F` cm$\"39,+++++!G$F17$F`cm$\"35-+++++gLF17$7$F`cm$\"3N,+++++?NF1Fajn7$Fe jn7$F`cm$\"3I-++++++OF17$7$F`cm$\"3c,+++++gPF1Fijn7$F][o7$F`cm$\"3_-++ +++SQF17$7$F`cm$\"3y,++++++SF1Fa[oFj_nF\\`n-%+AXESLABELSG6%Q\"x6\"Q\"y F\\\\o-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(Fd`m;Fdam$\"\"%F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv e 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Limit((x^2+2*x)/(x^2+x-2),x=-2);\n` `=simplify(%);\n``=value(rhs(%));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #-%&LimitG6$*&,&*$)%\"xG\"\"#\"\"\"F,*&F+F,F*F,F,F,,(F(F,F*F,F+!\"\"F/ /F*!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*&%\"xG\"\" \",&F)F*F*!\"\"F,/F)!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"\"# \"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 320 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 18 "Find the value of " }{XPPEDIT 18 0 "Limit((x^3+1)/(x+1),x = -1);" "6# -%&LimitG6$*&,&*$%\"xG\"\"$\"\"\"F+F+F+,&F)F+F+F+!\"\"/F),$F+F-" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 321 8 "Solution" }{TEXT -1 2 " : " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit((x^3+1)/( x+1),x = -1) = Limit((x+1)*(x^2-x+1)/(x+1),x = -1);" "6#/-%&LimitG6$*& ,&*$%\"xG\"\"$\"\"\"F,F,F,,&F*F,F,F,!\"\"/F*,$F,F.-F%6$*(,&F*F,F,F,F,, (*$F*\"\"#F,F*F.F,F,F,,&F*F,F,F,F./F*,$F,F." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Limit(``(x^2-x+1),x = \+ -1);" "6#/%!G-%&LimitG6$-F$6#,(*$%\"xG\"\"#\"\"\"F,!\"\"F.F./F,,$F.F/ " }{XPPEDIT 18 0 "`` = 3;" "6#/%!G\"\"$" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT 319 11 "Explanation" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 38 "The \"sum of two cubes\" factorisation " }{XPPEDIT 18 0 "x^3+1=(x+1)*(x^2-x+1)" "6#/,&*$%\"xG\"\"$\"\"\"F(F(*&,&F&F(F(F(F(,(*$ F&\"\"#F(F&!\"\"F(F(F(" }{TEXT -1 11 " shows that" }}{PARA 256 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "(x^3+1)/(x+1)=(x+1)*(x^2-x+1)/(x+1) " "6#/*&,&*$%\"xG\"\"$\"\"\"F)F)F),&F'F)F)F)!\"\"*(,&F'F)F)F)F),(*$F' \"\"#F)F'F+F)F)F),&F'F)F)F)F+" }{XPPEDIT 18 0 "``=x^2-x+1" "6#/%!G,(*$ %\"xG\"\"#\"\"\"F'!\"\"F)F)" }{TEXT -1 7 ", when " }{XPPEDIT 18 0 "x<> -1" "6#0%\"xG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "The graph of " }{XPPEDIT 18 0 "y=x^2-x+1" "6#/%\"yG,(*$%\"xG\" \"#\"\"\"F'!\"\"F)F)" }{TEXT -1 82 " has no breaks (anywhere) since it is a parabola with the same shape as the graph " }{XPPEDIT 18 0 "y=x^ 2" "6#/%\"yG*$%\"xG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "Hence the values of " }{XPPEDIT 18 0 "x^2-x+1" "6#,(*$%\"xG\"\"#\" \"\"F%!\"\"F'F'" }{TEXT -1 10 " approach " }{XPPEDIT 18 0 "(-1)^2-(-1) +1=3" "6#/,(*$,$\"\"\"!\"\"\"\"#F',$F'F(F(F'F'\"\"$" }{TEXT -1 4 " as \+ " }{TEXT 322 1 "x" }{TEXT -1 12 " approaches " }{XPPEDIT 18 0 "-1" "6# ,$\"\"\"!\"\"" }{TEXT -1 22 ". Hence the values of " }{XPPEDIT 18 0 "( x^3+1)/(x+1)" "6#*&,&*$%\"xG\"\"$\"\"\"F(F(F(,&F&F(F(F(!\"\"" }{TEXT -1 21 " also approach 3, as " }{TEXT 323 1 "x" }{TEXT -1 12 " approach es " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 "y=(x^3+1)/(x+1) " "6#/%\"yG*&,&*$%\"xG\"\"$\"\"\"F*F*F*,&F(F*F*F*!\"\"" }{TEXT -1 50 " is obtained by \"punching a hole\" in the parabola " }{XPPEDIT 18 0 " y=x^2-x+1" "6#/%\"yG,(*$%\"xG\"\"#\"\"\"F'!\"\"F)F)" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "``(-1,3)" "6#-%!G6$,$\"\"\"!\"\"\"\"$" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "f := x -> (x^3+1)/(x+1):\n'f(x)'=f(x);\nplot([f (x),[[-1,3]]$2],x=-2..3,y=0..7,style=[line,point$2],\n symbol=ci rcle,symbolsize=[15,18],color=[red,brown$2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&,&*$)F'\"\"$\"\"\"F-F-F-F-,&F'F-F-F-!\" \"" }}{PARA 13 "" 1 "" {GLPLOT2D 348 295 295 {PLOTDATA 2 "6(-%'CURVESG 6&7S7$$!\"#\"\"!$\"\"(F*7$$!3smm;HU,\"*=!#<$\"3Ipu2W!\\pY'F07$$!3SL$3F H'='z\"F0$\"3dS;#3\\rC-'F07$$!3gmmTgBa*o\"F0$\"3k()*e![d4WbF07$$!3amm \"H_\">#e\"F0$\"3CFg5Q:_&3&F07$$!3ML$3_!4Nv9F0$\"3+#Q2()>6?l%F07$$!3km ;/wfHw8F0$\"31],r*e'[qUF07$$!3;+]PM.tt7F0$\"3M]?3*H>h*QF07$$!3em;/,oln 6F0$\"3sC011#z5`$F07$$!3%)**\\(oWB>1\"F0$\"3izy*Q&[g*=$F07$$!3eJLLepjJ &*!#=$\"3!G*z)os%ohGF07$$!3Ulm;z/ot&)Fhn$\"3W,^],%H&>BJ6BF07$$!3A*****\\7)Q7kFhn$\"355k:FgU_?F07$$!3e**** *\\i^)o`Fhn$\"3P^E@S38D=F07$$!3vlmT50A@WFhn$\"3bc*e!fRfP;F07$$!3OKLLea R%H$Fhn$\"3-1A>g*pzV\"F07$$!3kJLLLo#)RBFhn$\"3-+KVz0t)G\"F07$$!3f***\\ PfO%H7Fhn$\"33;bv-)e!Q6F07$$!3MSLLL3`lC!#>$\"3AXBc#>j_-\"F07$$\"3+L+]i !f#=$)Ffq$\"3)3'3eFvOP#*Fhn7$$\"3+-+v=xpe=Fhn$\"3aW'=A+yn[)Fhn7$$\"3pw!41+vFhn7$$\"3!fL$e*)>pxgFhn$\"3a37W-?9;wFhn 7$$\"3w++v$f4t.(Fhn$\"3D!ey!QI1:zFhn7$$\"3OPL$e*Gst!)Fhn$\"3N3q.WsxW%) Fhn7$$\"3Y+++]#RW9*Fhn$\"3idSppwj<#*Fhn7$$\"3:++DJE>>5F0$\"3())HW$))4c >5F07$$\"3F+]i!RU07\"F0$\"3W!fi&eG2N6F07$$\"3+++v=S2L7F0$\"3Yvi'p^(R(G \"F07$$\"3Jmmm\"p)=M8F0$\"3QWXIt2(eW\"F07$$\"3B++](=]@W\"F0$\"3r&G1e(p kP;F07$$\"35L$e*[$z*R:F0$\"3J)zfi/d:$=F07$$\"3e++]iC$pk\"F0$\"3;:j`tSX l?F07$$\"3[m;H2qcZd*z DF07$$\"3Ymm;/Ogb>F0$\"3M*)HY_=yoGF07$$\"3w**\\ilAFj?F0$\"3_>BZw,#Q>$F 07$$\"3yLLL$)*pp;#F0$\"3(\\LM`#*)yGNF07$$\"3)RL$3xe,tAF0$\"3s$e*Q+`e$* QF07$$\"3Cn;HdO=yBF0$\"3%H]<0&QdxUF07$$\"3a+++D>#[Z#F0$\"3-32YN;#*\\YF 07$$\"3SnmT&G!e&e#F0$\"3Ao'=p7X'*4&F07$$\"3#RLLL)Qk%o#F0$\"3.]d)p*)oE_ &F07$$\"37+]iSjE!z#F0$\"3uG)**4\"*>`*fF07$$\"3a+]P40O\"*GF0$\"3+i#)z&3 0'okF07$$\"\"$F*F+-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*F_[l-%&STYLEG6# %%LINEG-%'SYMBOLG6$%(DEFAULTG\"#:-F$6&7#7$$!\"\"F*Ffz-Fiz6&F[[l$\")#)e qkF^[l$\"))eqk\"F^[lFc\\l-Fa[l6#%&POINTG-Fe[l6$Fg[l\"#=-F$6&F[\\lF_\\l Fe\\lFd[l-%+AXESLABELSG6$Q\"x6\"Q\"yFa]l-Fe[l6#%'CIRCLEG-%%VIEWG6$;F(F fz;F_[lF+" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cu rve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "Limit((x^3+1)/(x+1),x=-1);\n``=simp lify(%);\n``=value(rhs(%));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&Li mitG6$*&,&*$)%\"xG\"\"$\"\"\"F,F,F,F,,&F*F,F,F,!\"\"/F*F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,(*$)%\"xG\"\"#\"\"\"F-F+!\"\"F- F-/F+F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 5 " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 308 8 "Question" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 19 "Find the value of " } {XPPEDIT 18 0 "Limit((x^2+x-2)/(x^2-1),x = 1)" "6#-%&LimitG6$*&,(*$%\" xG\"\"#\"\"\"F)F+F*!\"\"F+,&*$F)F*F+F+F,F,/F)F+" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT 309 8 "Solution" }{TEXT -1 2 ": " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit((x^2+x-2)/(x^2-1),x = 1) = Limit((x-1)*(x+2)/((x-1)*(x+1)),x = 1);" "6#/-%&LimitG6$*&,(*$%\"xG\" \"#\"\"\"F*F,F+!\"\"F,,&*$F*F+F,F,F-F-/F*F,-F%6$*(,&F*F,F,F-F,,&F*F,F+ F,F,*&,&F*F,F,F-F,,&F*F,F,F,F,F-/F*F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Limit((x+2)/(x+1),x = 1);" "6 #/%!G-%&LimitG6$*&,&%\"xG\"\"\"\"\"#F+F+,&F*F+F+F+!\"\"/F*F+" } {XPPEDIT 18 0 "`` = 3/2;" "6#/%!G*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 15 ": The graph of " }{XPPEDIT 18 0 "y=(x^2+x-2)/(x^2-1)" "6#/%\"yG*&,(*$%\"xG\"\"#\"\" \"F(F*F)!\"\"F*,&*$F(F)F*F*F+F+" }{TEXT -1 51 " is obtained by \"punch ing a hole\" in the graph of " }{XPPEDIT 18 0 "y = (x+2)/(x+1);" "6#/ %\"yG*&,&%\"xG\"\"\"\"\"#F(F(,&F'F(F(F(!\"\"" }{TEXT -1 13 " at the po int" }{XPPEDIT 18 0 "``(1,3/2);" "6#-%!G6$\"\"\"*&\"\"$F&\"\"#!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 332 "f := x -> (x^2+x-2)/(x^2-1):\n'f(x)'=f(x);\np1 \+ := plot(f(x),x=-3..3,y=-2..4,discont=true,color=red):\np2 := plot([[[1 ,3/2]]$2],style=[point$2],\n symbol=[circle$2],symbolsize=[15,18 ],color=[brown$2]):\np3 := plots[implicitplot](\{x=-1,y=1\},x=-3..3,y= -2..4,\n color=COLOR(RGB,.4,.4,.4),linestyle=3):\nplots[display ]([p1,p2,p3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&,(*$ )F'\"\"#\"\"\"F-F'F-F,!\"\"F-,&F*F-F-F.F." }}{PARA 13 "" 1 "" {GLPLOT2D 348 295 295 {PLOTDATA 2 "6)-%'CURVESG6%7gn7$$!\"$\"\"!$\"3++ ++++++]!#=7$$!3s\\uz\"p0k&H!#<$\"3YNy::de))[F-7$$!3j4zKy!fe%QT%F-7$$!3W6^;\"R=0v#F1$\"35Z() G!\\1uG%F-7$$!3sB:iM@\\4FF1$\"3o[%R8W4.:%F-7$$!3]yaT@F1nEF1$\"3I#**f$e aU,SF-7$$!3#>pv)z$pZi#F1$\"3yMG(yO!GXQF-7$$!3oO&*ezaE\"e#F1$\"3%yG4rB^ fn$F-7$$!3=Dy`?s%Ha#F1$\"3=())z\"4p*)=NF-7$$!3Qq0]O*4)*\\#F1$\"3Ln6i* \\)[KLF-7$$!3I80jEb\\cCF1$\"3eXC&pz/U8$F-7$$!3nPdvmSv9CF1$\"3_TQbxOjJH 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**z%F-Fdam7$7$$!33#************f$F-FdamF`hm7$Fdhm7$$!3C$************R# F-Fdam7$7$$!3=#************>\"F-FdamFhhm7$F\\im7$$\"3CR4vZ\"Q8m'!#LFda m7$7$$\"3s2++++++7F-FdamF`im7$Feim7$$\"3c1++++++CF-Fdam7$7$$\"3i2+++++ +OF-FdamFiim7$F]jm7$$\"3[1++++++[F-Fdam7$7$$\"3a2++++++gF-FdamFajm7$Fe jm7$$\"3S1++++++sF-Fdam7$7$$\"3Y2++++++%)F-FdamFijm7$F][n7$$\"3I1+++++ +'*F-Fdam7$7$$\"3u++++++!3\"F1FdamFa[n7$Fe[n7$$\"3g+++++++7F1Fdam7$7$$ \"3t++++++?8F1FdamFi[n7$F]\\n7$$\"3h++++++S9F1Fdam7$7$$\"3s++++++g:F1F damFa\\n7$Fe\\n7$$\"3g++++++!o\"F1Fdam7$7$$\"3q+++++++=F1FdamFi\\n7$F] ]n7$$\"3#3++++++#>F1Fdam7$7$$\"3#4++++++/#F1FdamFa]n7$Fe]n7$$\"3/,++++ +g@F1Fdam7$7$$\"39,+++++!G#F1FdamFi]n7$F]^n7$$\"3C,++++++CF1Fdam7$7$$ \"3N,+++++?DF1FdamFa^n7$Fe^n7$$\"3Y,+++++SEF1Fdam7$7$$\"3c,+++++gFF1Fd amFi^n7$F]_n7$$\"3m,+++++!)GF1Fdam7$7$$\"3y,++++++IF1FdamFa_n-%&COLORG 6&F[am$\"\"%!\"\"F[`nF[`n-%*LINESTYLEG6#Fe`m-F$6V7$7$$F]`nF*$!\"#F*7$F e`n$!3>************R=F17$7$Fe`n$!3-++++++g6F17$7$Fe` n$!3/++++++S5F1F`bn7$Fdbn7$Fe`n$!3E#************z)F-7$7$Fe`n$!3U++++++ +!)F-Fhbn7$F\\cn7$Fe`n$!3O#************R'F-7$7$Fe`n$!3a+++++++cF-F`cn7 $Fdcn7$Fe`n$!3W#*************RF-7$7$Fe`n$!3i+++++++KF-Fhcn7$F\\dn7$Fe` n$!3a#************f\"F-7$7$Fe`n$!372++++++!)FgrF`dn7$Fddn7$Fe`n$\"3ut+ +++++!)Fgr7$7$Fe`n$\"3?*************f\"F-Fhdn7$F\\en7$Fe`n$\"3G2++++++ KF-7$7$Fe`n$\"37**************RF-F`en7$Fden7$Fe`n$\"3?2++++++cF-7$7$Fe `n$\"3-*************R'F-Fhen7$F\\fn7$Fe`n$\"332++++++!)F-7$7$Fe`n$\"3# *)************z)F-F`fn7$Fdfn7$Fe`n$\"3q++++++S5F17$7$Fe`n$\"3*)******* *****>6F1Fhfn7$F\\gn7$Fe`n$\"3q++++++!G\"F17$7$Fe`n$\"3))************f 8F1F`gn7$Fdgn7$Fe`n$\"3o++++++?:F17$7$Fe`n$\"3')*************f\"F1Fhgn 7$F\\hn7$Fe`n$\"3o++++++g++++++?$F17$7$Fe`n$\"39,+++++!G$F1F`[o7$Fd[o7$Fe`n$\"3<-+++++SM F17$7$Fe`n$\"3N,+++++?NF1Fh[o7$F\\\\o7$Fe`n$\"3Q-+++++!o$F17$7$Fe`n$\" 3c,+++++gPF1F`\\o7$Fd\\o7$Fe`n$\"3f-+++++?RF17$7$Fe`n$\"3y,++++++SF1Fh \\oFh_nF^`n-%+AXESLABELSG6%Q\"x6\"Q\"yFc]o-%%FONTG6#%(DEFAULTG-%%VIEWG 6$;F(Fd`m;Ff`n$F\\`nF*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Limit((x^2+x-2)/(x^2-1),x=1);\n``=simplify(%);\n``=value(rhs(%)); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&,(*$)%\"xG\"\"#\"\" \"F,F*F,F+!\"\"F,,&F(F,F,F-F-/F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%!G-%&LimitG6$*&,&%\"xG\"\"\"\"\"#F+F+,&F*F+F+F+!\"\"/F*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"\"$\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 6" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 " limit((x^5-1)/(x-1),x = 1);" "6#-%&limitG6$*&,&*$%\"xG\"\"&\"\"\"F+!\" \"F+,&F)F+F+F,F,/F)F+" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Limit((x^5-1)/(x-1),x=1);\n``=simplify(%) ;\n``=value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&, &*$)%\"xG\"\"&\"\"\"F,F,!\"\"F,,&F*F,F,F-F-/F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,,*$)%\"xG\"\"%\"\"\"F-*$)F+\"\"$F-F-*$) F+\"\"#F-F-F+F-F-F-/F+F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G\"\"& " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 7" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 324 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 19 "Find the value of " }{XPPEDIT 18 0 "Limit((2*x^3+3*x^2-23*x-12)/(4*x ^3-x),x = -1/2);" "6#-%&LimitG6$*&,**&\"\"#\"\"\"*$%\"xG\"\"$F*F**&F-F **$F,F)F*F**&\"#BF*F,F*!\"\"\"#7F2F*,&*&\"\"%F**$F,F-F*F*F,F2F2/F,,$*& F*F*F)F2F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 325 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 11 " The cubic " } {XPPEDIT 18 0 "``(2*x^3+3*x^2-23*x-12)" "6#-%!G6#,**&\"\"#\"\"\"*$%\"x G\"\"$F)F)*&F,F)*$F+F(F)F)*&\"#BF)F+F)!\"\"\"#7F1" }{TEXT -1 56 " can \+ be factored by first searching for a rational root " }{XPPEDIT 18 0 "x =p/q" "6#/%\"xG*&%\"pG\"\"\"%\"qG!\"\"" }{TEXT -1 9 " , where " } {TEXT 326 2 "p " }{TEXT -1 4 "and " }{TEXT 327 1 "q" }{TEXT -1 40 " ar e integers, and corresponding factor " }{XPPEDIT 18 0 "``(q*x-p)" "6#- %!G6#,&*&%\"qG\"\"\"%\"xGF)F)%\"pG!\"\"" }{TEXT -1 92 ". Trial an erro r can be used along with the fact that, if such a rational root exists , then " }{TEXT 329 1 "p" }{TEXT -1 33 " must divide exactly into12, a nd " }{TEXT 328 1 "q" }{TEXT -1 29 " must divide exactly into 2. " }} {PARA 0 "" 0 "" {TEXT -1 161 "When one factor has been found, it can b e divided into the cubic to obtain a residual quadratic factor. This q uadratic may then be factored in the standard way. " }}{PARA 0 "" 0 " " {TEXT -1 43 "This eventually leads to the factorisation " }{XPPEDIT 18 0 "2*x^3+3*x^2-23*x-12=(x+4)*(2*x+1)*(x-3)" "6#/,**&\"\"#\"\"\"*$% \"xG\"\"$F'F'*&F*F'*$F)F&F'F'*&\"#BF'F)F'!\"\"\"#7F/*(,&F)F'\"\"%F'F', &*&F&F'F)F'F'F'F'F',&F)F'F*F/F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 34 "The denominator factors easily as " }{XPPEDIT 18 0 "4*x^3 -x=x*(2*x-1)*(2*x+1)" "6#/,&*&\"\"%\"\"\"*$%\"xG\"\"$F'F'F)!\"\"*(F)F' ,&*&\"\"#F'F)F'F'F'F+F',&*&F/F'F)F'F'F'F'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 159 ": If this limit is similar to all the other examples, then you might hazard a guess that the numerator and denominator have a common factor which is zero when " }{XPPEDIT 18 0 "x=-1/2" "6#/%\"xG,$*&\"\" \"F'\"\"#!\"\"F)" }{TEXT -1 38 ". This would suggest the common factor " }{XPPEDIT 18 0 "``(2*x+1)" "6#-%!G6#,&*&\"\"#\"\"\"%\"xGF)F)F)F)" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "We now have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Limit((2*x^3+3*x^2-23*x-12)/(4*x^3-x),x = -1/2)=Limit(( x+4)*(2*x+1)*(x-3)/(x*(2*x-1)*(2*x+1)),x = -1/2)" "6#/-%&LimitG6$*&,** &\"\"#\"\"\"*$%\"xG\"\"$F+F+*&F.F+*$F-F*F+F+*&\"#BF+F-F+!\"\"\"#7F3F+, &*&\"\"%F+*$F-F.F+F+F-F3F3/F-,$*&F+F+F*F3F3-F%6$**,&F-F+F7F+F+,&*&F*F+ F-F+F+F+F+F+,&F-F+F.F3F+*(F-F+,&*&F*F+F-F+F+F+F3F+,&*&F*F+F-F+F+F+F+F+ F3/F-,$*&F+F+F*F3F3" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``=limit((x+4)*(x-3)/(x*(2*x-1)),x = -1/2)" "6#/%!G- %&limitG6$*(,&%\"xG\"\"\"\"\"%F+F+,&F*F+\"\"$!\"\"F+*&F*F+,&*&\"\"#F+F *F+F+F+F/F+F//F*,$*&F+F+F3F/F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(- 1/2+4)*(-1/2-3)/(``(-1/2)*(-2))" "6#/%!G*(,&*&\"\"\"F(\"\"#!\"\"F*\"\" %F(F(,&*&F(F(F)F*F*\"\"$F*F(*&-F$6#,$*&F(F(F)F*F*F(,$F)F*F(F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=``(7/2) *(-7/2)" "6#/%!G*&-F$6#*&\"\"(\"\"\"\"\"#!\"\"F*,$*&F)F*F+F,F,F*" } {XPPEDIT 18 0 "``=-49/4" "6#/%!G,$*&\"#\\\"\"\"\"\"%!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "f := x ->(2*x^3+3*x^2-23*x-12)/(4*x^3-x):\n'f(x)'=f( x);\n``=map(``,factor(numer(f(x))))/factor(denom(f(x)));\n``=eval(rhs( %),``=(_U->_U));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&,* *&\"\"#\"\"\")F'\"\"$F,F,*&F.F,)F'F+F,F,*&\"#BF,F'F,!\"\"\"#7F3F,,&*& \"\"%F,F-F,F,F'F3F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*.-F$6#,&% \"xG\"\"\"\"\"%F*F*-F$6#,&F*F**&\"\"#F*F)F*F*F*-F$6#,&F)F*\"\"$!\"\"F* F)F5,&*&F0F*F)F*F*F*F5F5F.F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G** ,&%\"xG\"\"\"\"\"%F(F(,&F'F(\"\"$!\"\"F(,&*&\"\"#F(F'F(F(F(F,F,F'F," } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Limit(f(x),x=-1/2);\n``=factor(%);\n``=value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&,**&\"\"#\"\"\")%\"xG\"\"$F*F** &F-F*)F,F)F*F**&\"#BF*F,F*!\"\"\"#7F2F*,&*&\"\"%F*F+F*F*F,F2F2/F,#F2F) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$**,&%\"xG\"\"\"\"\" %F+F+,&F*F+\"\"$!\"\"F+,&*&\"\"#F+F*F+F+F+F/F/F*F//F*#F/F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#!#\\\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 " Finding limits numerically" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 131 "First we set up two procedures fo r generating sequences of numbers which approach a given number from t he right and from the left.\n" }}{PARA 0 "" 0 "" {TEXT 303 22 "rightap proach(c, h, n)" }{TEXT -1 22 " generates a list of " }{TEXT 331 1 "n " }{TEXT -1 10 " numbers: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x[1] = c+h,x[2] = c+h^2,x[3] = c+h^3,` . . . `,x[n] = c +h^n" "6'/&%\"xG6#\"\"\",&%\"cGF'%\"hGF'/&F%6#\"\"#,&F)F'*$F*F.F'/&F%6 #\"\"$,&F)F'*$F*F4F'%(~.~.~.~G/&F%6#%\"nG,&F)F')F*F;F'" }{TEXT -1 3 " \+ , " }}{PARA 0 "" 0 "" {TEXT -1 13 " which, if " }{TEXT 332 1 "h" } {TEXT -1 62 " is a positive number less than 1, will approach the cons tant " }{TEXT 333 1 "c" }{TEXT -1 17 " from the right.\n" }}{PARA 0 " " 0 "" {TEXT 303 21 "leftapproach(c, h, n)" }{TEXT -1 22 " generates \+ a list of " }{TEXT 330 1 "n" }{TEXT -1 11 " numbers: " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x[1] = c-h,x[2] = c-h^2,x[3] = c -h^3,` . . . `,x[n] = c-h^n" "6'/&%\"xG6#\"\"\",&%\"cGF'%\"hG!\"\"/&F% 6#\"\"#,&F)F'*$F*F/F+/&F%6#\"\"$,&F)F'*$F*F5F+%(~.~.~.~G/&F%6#%\"nG,&F )F')F*F " 0 "" {MPLTEXT 1 0 458 "rightapproach := proc(c::realcons,h::realcons,n::posint)\n loca l i;\n if evalf(h<=0) or evalf(h>1) then\n error \"the 2nd argu ment must be a positive constant less than 1\"\n end if;\n return \+ [seq(c+h^i,i=1..n)];\nend proc:\n\nleftapproach := proc(c::constant,h: :constant,n::posint)\n local i;\n if evalf(h<=0) or evalf(h>1) the n\n error \"the 2nd argument must be a positive constant less tha n 1\";\n end if;\n return [seq(c-h^i,i=1..n)];\nend proc:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "A sequenc e of 5 numbers which approach 2 from the right starting with 2.1\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "rightapproach(2,0.1,5);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7'$\"#@!\"\"$\"$,#!\"#$\"%,?!\"$$\"&,+ #!\"%$\"',+?!\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "A sequence of 5 numbers which approach 2 from the left st arting with 1.9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "leftapproach(2,0.1,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'$\"#>!\"\"$\"$*>!\"#$\"%**>!\"$$\"&***>!\"%$\"'****>! \"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "W e shall make use of the Maple procedure " }{TEXT 0 3 "map" }{TEXT -1 108 " which can be used to apply a function to all the members of a li st.\nFor example, we can apply the function " }{XPPEDIT 18 0 "f(x) = x ^2;" "6#/-%\"fG6#%\"xG*$F'\"\"#" }{TEXT -1 13 " to the list " } {XPPEDIT 18 0 "[1,2,3,4]" "6#7&\"\"\"\"\"#\"\"$\"\"%" }{TEXT -1 13 " a s follows:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "f := x -> x^ 2;\nmap(f,[1,2,3,4,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#% \"xG6\"6$%)operatorG%&arrowGF(*$)9$\"\"#\"\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"\"\"\"%\"\"*\"#;\"#D" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 23 "The function f, wh ere " }{XPPEDIT 18 0 "f(x) = (x^3-8)/(x-2);" "6#/-%\"fG6#%\"xG*&,&*$F' \"\"$\"\"\"\"\")!\"\"F,,&F'F,\"\"#F.F." }{TEXT -1 107 " , has no value for x equal to 2. If we try to substitute x equal to 0, we get the m eaningless expression " }{XPPEDIT 18 0 "0/0;" "6#*&\"\"!\"\"\"F$!\"\" " }{TEXT -1 98 ".\nNevertheless it may happen that the values of f(x) \+ approach some fixed number as x approaches 2." }}{PARA 0 "" 0 "" {TEXT -1 24 "In order to investigate " }{XPPEDIT 18 0 "limit((x^3-8)/( x-2),x = 2);" "6#-%&limitG6$*&,&*$%\"xG\"\"$\"\"\"\"\")!\"\"F+,&F)F+\" \"#F-F-/F)F/" }{TEXT -1 29 " , we shall obtain values of " }{XPPEDIT 18 0 "(x^3-8)/(x-2);" "6#*&,&*$%\"xG\"\"$\"\"\"\"\")!\"\"F(,&F&F(\"\"# F*F*" }{TEXT -1 144 " corresponding to values of x approaching 2 from the right and from the left.\n\nFirst we set up lists of 7 numbers ap proaching 2 from each side:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "( Make sure that you have activated the procedu res " }{TEXT 0 13 "rightapproach" }{TEXT -1 5 " and " }{TEXT 0 12 "le ftapproach" }{TEXT -1 41 " defined above. )\n\nWe apply the function \+ " }{XPPEDIT 18 0 "f(x)=(x^3-8)/(x-2)" "6#/-%\"fG6#%\"xG*&,&*$F'\"\"$\" \"\"\"\")!\"\"F,,&F'F,\"\"#F.F." }{TEXT -1 71 " to lists of numbers wh ich approach 2 from the right and from the left." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "f := x->(x^3 -8)/(x-2);\nrightapproach(2,0.1,7);\nmap(f,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,&*$)9$\"\"$ \"\"\"F2\"\")!\"\"F2,&F0F2\"\"#F4F4F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)$\"#@!\"\"$\"$,#!\"#$\"%,?!\"$$\"&,+#!\"%$\"',+?!\"&$ \"(,++#!\"'$\"),++?!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)$\"++++h7 !\")$\"+++,17F&$\"++,g+7F&$\"+++1+7F&$\"+++,+7F&$\"+++++7F&F/" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "f := x->(x^3-8)/(x-2);\nleft approach(2,0.1,7);\nmap(f,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"f Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,&*$)9$\"\"$\"\"\"F2\"\")!\"\"F2 ,&F0F2\"\"#F4F4F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)$\"#>!\"\"$ \"$*>!\"#$\"%**>!\"$$\"&***>!\"%$\"'****>!\"&$\"(*****>!\"'$\")******> !\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)$\"++++T6!\")$\"+++,%>\"F&$ \"++,S*>\"F&$\"+++%**>\"F&$\"+++***>\"F&$\"+++++7F&F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "It looks as though" }{XPPEDIT 18 0 "limit((x^3-8)/(x-2),x = 2) = 12;" "6#/-%&limitG6$*&,&* $%\"xG\"\"$\"\"\"\"\")!\"\"F,,&F*F,\"\"#F.F./F*F0\"#7" }{TEXT -1 2 ". \n" }}{PARA 0 "" 0 "" {TEXT -1 160 "There is a significant loss of pre cision in the last three members of each list of values due to the sub traction of nearly equal numbers during the calculation." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "To be more confide nt about our value for the limit, we could increase the number of digi ts used." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "Digits := 15:\nf := x->(x^3-8)/(x-2);\nrightapproach( 2,0.1,7);\nmap(f,%):\nevalf(%,10);\nDigits := 10:\n" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,&*$)9$\" \"$\"\"\"F2\"\")!\"\"F2,&F0F2\"\"#F4F4F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)$\"#@!\"\"$\"$,#!\"#$\"%,?!\"$$\"&,+#!\"%$\"',+?!\"&$ \"(,++#!\"'$\"),++?!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)$\"++++h7 !\")$\"+++,17F&$\"++,g+7F&$\"+,+1+7F&$\"++g++7F&$\"++1++7F&$\"+g+++7F& " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 9 "Consider " }{XPPEDIT 18 0 "limit((sqrt(x )-2)/(x-4),x = 4);" "6#-%&limitG6$*&,&-%%sqrtG6#%\"xG\"\"\"\"\"#!\"\"F ,,&F+F,\"\"%F.F./F+F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "f := x->(sqrt(x)-2)/(x-4 );\nrightapproach(4,0.1,5);\nmap(f,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,&-%%sqrtG6#9$\"\"\" \"\"#!\"\"F2,&F1F2\"\"%F4F4F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7 '$\"#T!\"\"$\"$,%!\"#$\"%,S!\"$$\"&,+%!\"%$\"',+S!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'$\"++tc%[#!#5$\"+++W)\\#F&$\"+++%)*\\#F&$\"+++++ DF&F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "f := x->(sqrt(x)-2)/(x-4);\nleftapproach(4,0.1,5);\nm ap(f,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)oper atorG%&arrowGF(*&,&-%%sqrtG6#9$\"\"\"\"\"#!\"\"F2,&F1F2\"\"%F4F4F(F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'$\"#R!\"\"$\"$*R!\"#$\"%**R!\"$$ \"&***R!\"%$\"'****R!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'$\"++M#e ^#!#5$\"++Sc,DF&$\"+++;+DF&$\"+++++DF&F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "It looks as though " }{XPPEDIT 18 0 "limit((sqrt(x)-2)/(x-4),x = 4) = 1/4;" "6#/-%&limitG6$*&,&-%%sqrtG6 #%\"xG\"\"\"\"\"#!\"\"F-,&F,F-\"\"%F/F//F,F1*&F-F-F1F/" }{TEXT -1 2 ". \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 9 "Consider \+ " }{XPPEDIT 18 0 "limit((1+x)^(1/x),x = 0);" "6#-%&limitG6$),&\"\"\"F( %\"xGF(*&F(F(F)!\"\"/F)\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "f := x->(1+x)^(1/x );\nrightapproach(0,0.1,9);\nmap(f,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(),&\"\"\"F.9$F.*&F.F.F/ !\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+$\"\"\"!\"\"$F%!\"#$F %!\"$$F%!\"%$F%!\"&$F%!\"'$F%!\"($F%!\")$F%!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+$\"+gCu$f#!\"*$\"+HQ\"[q#F&$\"+KR#pr#F&$\"+Ff9=FF&$\" +P#o#=FF&$\"+p/G=FF&$\"+#p\"G=FF&$\"+:=G=FF&$\"+F=G=FF&" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "f := \+ x->(1+x)^(1/x);\nleftapproach(0,0.1,9);\nmap(f,%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(),&\"\"\"F.9 $F.*&F.F.F/!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+$!\"\"F%$F %!\"#$F%!\"$$F%!\"%$F%!\"&$F%!\"'$F%!\"($F%!\")$F%!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+$\"+\"*>(z'G!\"*$\"+E!**>t#F&$\"+;Ak>FF&$\"+bxT= FF&$\"+?aH=FF&$\"+)=$G=FF&$\"+k>G=FF&$\"+U=G=FF&$\"+I=G=FF&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "It looks as tho ugh " }{XPPEDIT 18 0 "Limit((1+x)^(1/x),x = 0);" "6#-%&LimitG6$),&\"\" \"F(%\"xGF(*&F(F(F)!\"\"/F)\"\"!" }{TEXT -1 30 " is approximately 2.71 828183. " }}{PARA 0 "" 0 "" {TEXT -1 48 "In fact the value of this lim it is the constant " }{XPPEDIT 18 0 "exp(1);" "6#-%$expG6#\"\"\"" } {TEXT -1 56 ", which is the base of natural logarithms. The function \+ " }{XPPEDIT 18 0 "exp(x);" "6#-%$expG6#%\"xG" }{TEXT -1 32 " is obtain ed in Maple by typing " }{TEXT 0 3 "exp" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(exp(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+G=G=F!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "Using t he Maple procedure " }{TEXT 0 5 "limit" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 80 "The examples of the previous sections can be worked \+ out using the Maple command " }{TEXT 0 5 "limit" }{TEXT -1 1 "." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit((x^3-8)/(x-2),x = 2);" "6#-%&LimitG6$*&,&*$%\"xG\"\"$\"\" \"\"\")!\"\"F+,&F)F+\"\"#F-F-/F)F/" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "limit((x^3-8)/(x-2),x=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Using " }{TEXT 0 5 "Limit" }{TEXT -1 9 " with an " } {TEXT 259 9 "uppercase" }{TEXT -1 31 " L just prints the limit in an \+ " }{TEXT 259 11 "unevaluated" }{TEXT -1 7 " form. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "The command " }{TEXT 0 5 "value" }{TEXT -1 51 " can then be used to obtain the value of the lim it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Limit((x^3-8)/(x-2),x=2);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&,&*$)%\"xG\"\"$\"\"\"F,\"\")!\"\"F,,& F*F,\"\"#F.F./F*F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Limit((sqrt(x)-2)/(x-4),x = 4);" "6#- %&LimitG6$*&,&-%%sqrtG6#%\"xG\"\"\"\"\"#!\"\"F,,&F+F,\"\"%F.F./F+F0" } {TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Limit((sqr t(x)-2)/(x-4),x=4);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&L imitG6$*&,&*$-%%sqrtG6#%\"xG\"\"\"F-\"\"#!\"\"F-,&F,F-\"\"%F/F//F,F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Limit((x^2+x-2)/(x^2-1),x = 1);" "6#-%&LimitG6$* &,(*$%\"xG\"\"#\"\"\"F)F+F*!\"\"F+,&*$F)F*F+F+F,F,/F)F+" }{TEXT -1 1 " \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Limit((x^2+x-2)/(x^2-1 ),x=1);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&,(* $)%\"xG\"\"#\"\"\"F,F*F,F+!\"\"F,,&F(F,F,F-F-/F*F," }}{PARA 11 "" 1 " " {XPPMATH 20 "6##\"\"$\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Limit((x^5-1)/(x-1),x = 1);" "6#-%&LimitG6$*&,&*$%\"xG\"\"&\"\" \"F+!\"\"F+,&F)F+F+F,F,/F)F+" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Limit((x^5-1)/(x-1),x=1);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&,&*$)%\"xG\"\"&\"\"\"F,F,!\"\"F ,,&F*F,F,F-F-/F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Limit((2*x^3+3*x^2-23*x-12)/(4*x^3-x) ,x = -1/2);" "6#-%&LimitG6$*&,**&\"\"#\"\"\"*$%\"xG\"\"$F*F**&F-F**$F, F)F*F**&\"#BF*F,F*!\"\"\"#7F2F*,&*&\"\"%F**$F,F-F*F*F,F2F2/F,,$*&F*F*F )F2F2" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Li mit((2*x^3+3*x^2-23*x-12)/(4*x^3-x),x=-1/2);\nvalue(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%&LimitG6$*&,**$)%\"xG\"\"$\"\"\"\"\"#*&F+F,)F* F-F,F,*&\"#BF,F*F,!\"\"\"#7F2F,,&F(\"\"%F*F2F2/F*#F2F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!#\\\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 6" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Limit((1+x)^(1/x),x = 0);" "6#-%&LimitG6$),&\"\"\"F(%\"xGF(*&F(F (F)!\"\"/F)\"\"!" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Limit((1+x)^(1/x),x=0);\nvalue(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%&LimitG6$),&\"\"\"F(%\"xGF(*&F(F(F)!\"\"/F)\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 32 "Graphical illustration of limits" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examp le 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 37 "In order to illustrate the fact that " }{XPPEDIT 18 0 " limit((x^3-8)/(x-2),x = 2) = 12;" "6#/-%&limitG6$*&,&*$%\"xG\"\"$\"\" \"\"\")!\"\"F,,&F*F,\"\"#F.F./F*F0\"#7" }{TEXT -1 37 ", we could try p lotting the graph of " }{XPPEDIT 18 0 "y = (x^3-8)/(x-2);" "6#/%\"yG*& ,&*$%\"xG\"\"$\"\"\"\"\")!\"\"F*,&F(F*\"\"#F,F," }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 31 "In a Maple plot, the graph of " } {XPPEDIT 18 0 "y = (x^3-8)/(x-2);" "6#/%\"yG*&,&*$%\"xG\"\"$\"\"\"\"\" )!\"\"F*,&F(F*\"\"#F,F," }{TEXT -1 49 " appears to be exactly the sam e as the parabola " }{XPPEDIT 18 0 "y = x^2+2*x+4;" "6#/%\"yG,(*$%\"xG \"\"#\"\"\"*&F(F)F'F)F)\"\"%F)" }{TEXT -1 18 ", where the factor" } {XPPEDIT 18 0 " ``(x-2)" "6#-%!G6#,&%\"xG\"\"\"\"\"#!\"\"" }{TEXT -1 110 " has been cancelled at the top and bottom. Maple ignores the fact that there is no value of the function at 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot((x^3-8) /(x-2),x=-3..2.2,y=0..13.4,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 276 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7S7$$!\"$\"\"!$\"\"(F*7$$!3ELLL )zam)G!#<$\"3i*=u!GjYflF07$$!3_mmTWP.)y#F0$\"3a&os6nkq>'F07$$!3[LL$[0C rn#F0$\"3d-1I&4XF\"eF07$$!3GLL$Q=zac#F0$\"3Uza,v]s]aF07$$!3%om;9%\\OaC F0$\"3//>#GQx^6&F07$$!3MLL3:yM^BF0$\"3A*\\It\"49E[F07$$!37++vZ&zYC#F0$ \"3/XrewrA\\XF07$$!3F0$\"3X_??3*\\/$QF07$$!3>LL$)pFm6=F0$\"3K =N$>X'zeOF07$$!3'******4$e]*p\"F0$\"3)4+.wS3$*[$F07$$!33+++l$))oe\"F0$ \"36tB(H&zVWLF07$$!3!*******o0Oy9F0$\"3m.S(R$)G)GKF07$$!3MLL3Lp!)z8F0$ \"3pn\"=kI`U9$F07$$!3ommmFrhi7F0$\"3#**)Qubx'*oIF07$$!3mmmm!*>Mj6F0$\" 39\\\\\"fg!oEIF07$$!3)****\\dShy/\"F0$\"3'eOgTr!H-IF07$$!3Yomm1_Tc%*!# =$\"3eKaFW[&H+$F07$$!35(***\\d5!\\L)Feq$\"3QYJ)[aDx-$F07$$!3K'***\\sV& pE(Feq$\"3!*o*>SQ&puIF07$$!3[kmmTSm_hFeq$\"3yYGwR*>![JF07$$!3qkm;uwRH^ Feq$\"3;JT;qwAPKF07$$!3:JLLemmDSFeq$\"3E+Nx)eEpN$F07$$!3QJL$3L+#zGFeq$ \"3nTSGzy02NF07$$!3M,+]A!)>\")=Feq$\"3iy&)\\b%\\\"fOF07$$!3j)HLL)=GL!) !#>$\"35V]6CxyXQF07$$\"3\\*******>o@5$F^t$\"3;4Tv3d+jSF07$$\"3q,++lLg* R\"Feq$\"3Y$>$zo&4&*H%F07$$\"3Y****\\i3k`CFeq$\"3#yH@tqJ4b%F07$$\"3g,+ +&zpRi$Feq$\"3wB2vHb7c[F07$$\"3MKLL$Ribn%Feq$\"3G'>1cJ@P:&F07$$\"3'Q++ +&>O)z&Feq$\"3kz?..C)e\\&F07$$\"3Hkm;H_y:oFeq$\"3#yfM(GjqFeF07$$\"3Y/+ +5w4GzFeq$\"3tyo8Ro;9iF07$$\"33IL$e(opu*)Feq$\"3V\"f(H:7R+mF07$$\"3[++ Dn%po+\"F0$\"3T+.ee]_FqF07$$\"3kLLL[x#Q6\"F0$\"3yrRi\\xEouF07$$\"3e*** \\i:.eA\"F0$\"3cgCP!p*>azF07$$\"3vmmm#y[OL\"F0$\"3k;;%3k;fW)F07$$\"3=n m;7l$RW\"F0$\"3Au8]vc#G(*)F07$$\"3_LLe.5J`:F0$\"3.J(>53(R>&*F07$$\"3I+ ++-[\"Ql\"F0$\"3,I9L+LF/5!#;7$$\"3CLL$o\\.!pF0$\"3KRP)3/f\"*=\"Fay 7$$\"3Y++vH\\,(3#F0$\"3cR*\\w6mHD\"Fay7$$\"3;+++++++AF0$\"3Q++++++C8Fa y-%+AXESLABELSG6$Q\"x6\"Q\"yF_[l-%'COLOURG6&%$RGBG$F*F*Fe[l$\"*++++\"! \")-%%VIEWG6$;F($\"#A!\"\";Fe[l$\"$M\"F_\\l" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 78 " \nWe could try to indicate that there is a point missing from \+ the graph where " }{XPPEDIT 18 0 "x = 2" "6#/%\"xG\"\"#" }{TEXT -1 40 " by plotting an open circle at the point" }{XPPEDIT 18 0 " ``(2,12)" "6#-%!G6$\"\"#\"#7" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "parabola := plot((x^3-8)/(x-2),x=-3..2.2,color=blue) :\nmissingpoint := plot([[2,12]],style=point,symbol=circle,color=blue) :\nplots[display]([parabola,missingpoint],view=[-3..2.2,0..13.4]);" }} {PARA 13 "" 1 "" {GLPLOT2D 305 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7S7$ $!\"$\"\"!$\"\"(F*7$$!3ELLL)zam)G!#<$\"3i*=u!GjYflF07$$!3_mmTWP.)y#F0$ \"3a&os6nkq>'F07$$!3[LL$[0Crn#F0$\"3d-1I&4XF\"eF07$$!3GLL$Q=zac#F0$\"3 Uza,v]s]aF07$$!3%om;9%\\OaCF0$\"3//>#GQx^6&F07$$!3MLL3:yM^BF0$\"3A*\\I t\"49E[F07$$!37++vZ&zYC#F0$\"3/XrewrA\\XF07$$!3F0$\"3X_??3*\\/ $QF07$$!3>LL$)pFm6=F0$\"3K=N$>X'zeOF07$$!3'******4$e]*p\"F0$\"3)4+.wS3 $*[$F07$$!33+++l$))oe\"F0$\"36tB(H&zVWLF07$$!3!*******o0Oy9F0$\"3m.S(R $)G)GKF07$$!3MLL3Lp!)z8F0$\"3pn\"=kI`U9$F07$$!3ommmFrhi7F0$\"3#**)Qubx '*oIF07$$!3mmmm!*>Mj6F0$\"39\\\\\"fg!oEIF07$$!3)****\\dShy/\"F0$\"3'eO gTr!H-IF07$$!3Yomm1_Tc%*!#=$\"3eKaFW[&H+$F07$$!35(***\\d5!\\L)Feq$\"3Q YJ)[aDx-$F07$$!3K'***\\sV&pE(Feq$\"3!*o*>SQ&puIF07$$!3[kmmTSm_hFeq$\"3 yYGwR*>![JF07$$!3qkm;uwRH^Feq$\"3;JT;qwAPKF07$$!3:JLLemmDSFeq$\"3E+Nx) eEpN$F07$$!3QJL$3L+#zGFeq$\"3nTSGzy02NF07$$!3M,+]A!)>\")=Feq$\"3iy&)\\ b%\\\"fOF07$$!3j)HLL)=GL!)!#>$\"35V]6CxyXQF07$$\"3\\*******>o@5$F^t$\" 3;4Tv3d+jSF07$$\"3q,++lLg*R\"Feq$\"3Y$>$zo&4&*H%F07$$\"3Y****\\i3k`CFe q$\"3#yH@tqJ4b%F07$$\"3g,++&zpRi$Feq$\"3wB2vHb7c[F07$$\"3MKLL$Ribn%Feq $\"3G'>1cJ@P:&F07$$\"3'Q+++&>O)z&Feq$\"3kz?..C)e\\&F07$$\"3Hkm;H_y:oFe q$\"3#yfM(GjqFeF07$$\"3Y/++5w4GzFeq$\"3tyo8Ro;9iF07$$\"33IL$e(opu*)Feq $\"3V\"f(H:7R+mF07$$\"3[++Dn%po+\"F0$\"3T+.ee]_FqF07$$\"3kLLL[x#Q6\"F0 $\"3yrRi\\xEouF07$$\"3e***\\i:.eA\"F0$\"3cgCP!p*>azF07$$\"3vmmm#y[OL\" F0$\"3k;;%3k;fW)F07$$\"3=nm;7l$RW\"F0$\"3Au8]vc#G(*)F07$$\"3_LLe.5J`:F 0$\"3.J(>53(R>&*F07$$\"3I+++-[\"Ql\"F0$\"3,I9L+LF/5!#;7$$\"3CLL$o\\.!p F0$\"3KRP)3/f\"*=\"Fay7$$\"3Y++vH\\,(3#F0$\"3cR*\\w6mHD\"Fay7$$\"3 ;+++++++AF0$\"3Q++++++C8Fay-%'COLOURG6&%$RGBG$F*F*F_[l$\"*++++\"!\")-F $6&7#7$$\"\"#F*$\"#7F*F[[l-%'SYMBOLG6#%'CIRCLEG-%&STYLEG6#%&POINTG-%+A XESLABELSG6%Q\"x6\"Q!Fg\\l-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F($\"#A!\"\"; F_[l$\"$M\"Fc]l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 102 "One way to \"check\" the limit is to zoom in towa rds the \"missing point\" so as to plot over an interval " }{XPPEDIT 18 0 "[2-delta,2+delta]" "6#7$,&\"\"#\"\"\"%&deltaG!\"\",&F%F&F'F&" } {TEXT -1 7 " where " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT -1 34 " is a small positive real number.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "delta := 0.001;\nparabola := plot((x^3-8)/(x-2),x=2- delta..2+delta,color=blue):\nmissingpoint := plot([[2,12]],style=point ,symbol=circle,color=blue):\nplots[display]([parabola,missingpoint]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&deltaG$\"\"\"!\"$" }}{PARA 13 " " 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$\"35+++++ +**>!#<$\"3A8+++,S*>\"!#;7$$\"3ML$3VfV!**>F*$\"3_\"=c![iU*>\"F-7$$\"3% o\"H[D:3**>F*$\"352$\\'***[%*>\"F-7$$\"3]Le0$=C\"**>F*$\"3=g!Rleu%*>\" F-7$$\"3YL3RBr;**>F*$\"3!p(GSV.]*>\"F-7$$\"3'o\"zjf)4#**>F*$\"3C3\\@yf _*>\"F-7$$\"3Q$e4;[\\#**>F*$\"3+;NHX(\\&*>\"F-7$$\"3=]i'y]!H**>F*$\"3S Ww0bVd*>\"F-7$$\"3Q$ezs$HL**>F*$\"3=o]'o!)*f*>\"F-7$$\"3+]7iI_P**>F*$ \"3S7kSx^i*>\"F-7$$\"3um;_M(=%**>F*$\"39!)**\\u7l*>\"F-7$$\"3IL3y_qX** >F*$\"3))Qw9hUn*>\"F-7$$\"3-+]1!>+&**>F*$\"3'=/?!R,q*>\"F-7$$\"31+]Z/N a**>F*$\"3CBQ_Bhs*>\"F-7$$\"33+]$fC&e**>F*$\"3#y=jZ;^(*>\"F-7$$\"3W$ez 6:B'**>F*$\"3yf$4\\!Rx*>\"F-7$$\"3ym;=C#o'**>F*$\"3+5m^X4!)*>\"F-7$$\" 3ymm#pS1(**>F*$\"3)plvF&Q#)*>\"F-7$$\"3#*\\i`A3v**>F*$\"3kx1t*\\])*>\" F-7$$\"3qmm(y8!z**>F*$\"3!QMIr3u)*>\"F-7$$\"3-]i.tK$)**>F*$\"31^9gm*** )*>\"F-7$$\"3'*\\(3zMu)**>F*$\"3w)QC.hC**>\"F-7$$\"3sm\"H_?<***>F*$\"3 WRI#QK]**>\"F-7$$\"3o;zihl&***>F*$\"3!e[lr$R(**>\"F-7$$\"3OL3#G,*****> F*$\"3A0'=xS****>\"F-7$$\"3F$ezw5V++#F*$\"3*>S$zke-+7F-7$$\"3**\\PQ#\\ \"3+?F*$\"3T]_4'*)[+?\"F-7$$\"3?Le\"*[H7+?F*$\"3n?<'3xt+?\"F-7$$\"3\"* ***pvxl,+#F*$\"379+Hp%*4+7F-7$$\"3\"**\\_qn2-+#F*$\"3#Q:W0hC,?\"F-7$$ \"3))\\i&p@[-+#F*$\"3zR[LO*[,?\"F-7$$\"3y*\\2'HKH+?F*$\"3qoDOYf<+7F-7$ $\"3_mmZvOL+?F*$\"3\\!)*>k@+-?\"F-7$$\"3#)**\\2goP+?F*$\"3n&eZ-8E-?\"F -7$$\"3=$eR<*fT+?F*$\"3q:&[Bh\\-?\"F-7$$\"33+])Hxe/+#F*$\"3#\\DQ[Gv-? \"F-7$$\"3!o\"H!o-*\\+?F*$\"3A4W)4W*H+7F-7$$\"3$*\\7k.6a+?F*$\"3J=TY\" pC.?\"F-7$$\"3um;WTAe+?F*$\"3:6bwy$\\.?\"F-7$$\"3z\\i!*3`i+?F*$\"3#H#[ WC_P+7F-7$$\"33LL*zym1+#F*$\"3!*RlD<,S+7F-7$$\"3QL3N1#42+#F*$\"3*f(z5u bU+7F-7$$\"3[;HYt7v+?F*$\"3/&)*=0#3X+7F-7$$\"3t***p(G**y+?F*$\"3X>2m>S Z+7F-7$$\"3ymT6KU$3+#F*$\"3=!yi)31]+7F-7$$\"38LLbdQ(3+#F*$\"3pwY*3RC0? \"F-7$$\"3u\\i`1h\"4+#F*$\"3\"y'o%yu\\0?\"F-7$$\"3-]P?Wl&4+#F*$\"3H))* >!=Sd+7F-7$$\"3*)***********4+#F*$\"3^9*****4+1?\"F--%'COLOURG6&%$RGBG $\"\"!F][lF\\[l$\"*++++\"!\")-F$6&7#7$$\"\"#F][l$\"#7F][lFhz-%&STYLEG6 #%&POINTG-%'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6%Q\"x6\"Q!6\"%(DEFAULTG-% %VIEWG6$;$\"%**>!\"$$\"%,?F_]lFh\\l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 189 "delta := 0.0001; \nparabola := plot((x^3-8)/(x-2),x=2-delta..2+delta,color=blue):\nmiss ingpoint := plot([[2,12]],style=point,symbol=circle,color=blue):\nplot s[display]([parabola,missingpoint]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%&deltaG$\"\"\"!\"%" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$\"3++++++!***>!#<$\"3Ql&**4+S**>\"!#;7 $$\"3WL3VfV!***>F*$\"3uNKd;E%**>\"F-7$$\"3!=H[D:3***>F*$\"3QLFP#*[%**> \"F-7$$\"3U$e0$=C\"***>F*$\"3953v^u%**>\"F-7$$\"3I$3RBr;***>F*$\"3!>H( 4G+&**>\"F-7$$\"3k\"zjf)4#***>F*$\"3=uI?#f_**>\"F-7$$\"3We4;[\\#***>F* $\"3'>\"*f%p\\&**>\"F-7$$\"3)\\i'y]!H***>F*$\"3MG`(4Vd**>\"F-7$$\"3Oez s$HL***>F*$\"3A/@ow*f**>\"F-7$$\"3-D@1Bv$***>F*$\"3?-uA9D'**>\"F-7$$\" 3ym@Xt=%***>F*$\"3)*p\"4W7l**>\"F-7$$\"3O$3y_qX***>F*$\"3sMCYBu'**>\"F -7$$\"3/+l+>+&***>F*$\"3/$Q`;,q**>\"F-7$$\"39+vW]V&***>F*$\"3%)okZ5E(* *>\"F-7$$\"35+NfC&e***>F*$\"3WPz#\\6v**>\"F-7$$\"3Sez6:B'***>F*$\"33e; @*Qx**>\"F-7$$\"3#o;=C#o'***>F*$\"3o*pgN4!)**>\"F-7$$\"3smEpS1(***>F*$ \"3!3a,XQ#)**>\"F-7$$\"33DOD#3v***>F*$\"3oLZT\\])**>\"F-7$$\"3umwy8!z* **>F*$\"3spqJ3u)**>\"F-7$$\"3;DOIFL)***>F*$\"3ck!3k***)**>\"F-7$$\"3/v 3zMu)***>F*$\"3cCQ*3Y#***>\"F-7$$\"3u;H_?<****>F*$\"3_&>BB.&***>\"F-7$ $\"3%=zihl&****>F*$\"3/#y*p$R(***>\"F-7$$\"3]$3#G,******>F*$\"3?4*=1%* ****>\"F-7$$\"3fezw5V++?F*$\"3%y(pY'e-+?\"F-7$$\"3Av$Q#\\\"3++#F*$\"31 Ywb*)[++7F-7$$\"3]$e\"*[H7++#F*$\"3B&4_pP2+?\"F-7$$\"3/+qvxl,+?F*$\"3T *H$oY*4+?\"F-7$$\"3<]_qn2-+?F*$\"3H7$o1Y7+?\"F-7$$\"37Dcp@[-+?F*$\"3h4 i2$*[,+7F-7$$\"3?]2'HKH++#F*$\"3P9='Qf<+?\"F-7$$\"3qmwanL.+?F*$\"3(\\D S1-?+?\"F-7$$\"3)**\\2goP++#F*$\"3xVju6E-+7F-7$$\"3TeR<*fT++#F*$\"3XSc nf\\-+7F-7$$\"39+&)Hxe/+?F*$\"3W)R\"fEv-+7F-7$$\"3/#H!o-*\\++#F*$\"3$Q re=%*H+?\"F-7$$\"3&\\7k.6a++#F*$\"3SBD^mC.+7F-7$$\"3!p;WTAe++#F*$\"3() eX#[$\\.+7F-7$$\"3-D1*3`i++#F*$\"3tOf#*=v.+7F-7$$\"3`L$*zym1+?F*$\"3YP Os2+/+7F-7$$\"3H$3N1#42+?F*$\"3=KK)GbU+?\"F-7$$\"3+#HYt7v++#F*$\"36!4s p2X+?\"F-7$$\"3(***p(G**y++#F*$\"3`p'\\jRZ+?\"F-7$$\"33<9@BM3+?F*$\"3n *pAY0]+?\"F-7$$\"3iL`v&Q(3+?F*$\"3z!\"%$\"&,+#Fa]lF j\\l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example \+ 2" }}{PARA 0 "" 0 "" {TEXT -1 40 "Similarly, we can illustrate the lim it " }{XPPEDIT 18 0 "limit((sqrt(x)-2)/(x-4),x = 4) = 1/4;" "6#/-%&li mitG6$*&,&-%%sqrtG6#%\"xG\"\"\"\"\"#!\"\"F-,&F,F-\"\"%F/F//F,F1*&F-F-F 1F/" }{TEXT -1 25 " by drawing the graph of " }{XPPEDIT 18 0 "y = (sqr t(x)-2)/(x-4);" "6#/%\"yG*&,&-%%sqrtG6#%\"xG\"\"\"\"\"#!\"\"F+,&F*F+\" \"%F-F-" }{TEXT -1 34 " and putting a \"hole\" at the point" } {XPPEDIT 18 0 "``(4,1/4);" "6#-%!G6$\"\"%*&\"\"\"F(F&!\"\"" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "curve := plot((sq rt(x)-2)/(x-4),x=0..5,color=blue):\nmissingpoint := plot([[4,0.25]],st yle=point,symbol=circle,color=blue):\nplots[display]([curve,missingpoi nt],view=[0..5,0..0.6]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6&7#7$$\"\"%\"\"!$\"3++++++++D!#=-%'COLOURG6& %$RGBG$F*F*F2$\"*++++\"!\")-%&STYLEG6#%&POINTG-%'SYMBOLG6#%'CIRCLEG-F$ 6$7en7$F2$\"3++++++++]F-7$$\"3/LL3x1h6o!#?$\"3o(fb(pe%=![F-7$$\"3gmmTN @Ki8!#>$\"326t0MOHCZF-7$$\"3#***\\7.K[V?FM$\"3[s'>mYjkm%F-7$$\"3ALL$3F WYs#FM$\"3\\NY7[!*z=YF-7$$\"3%)***\\iSmp3%FM$\"3ke\\q?5*4a%F-7$$\"3Wmm mT&)G\\aFM$\"3+%3]L)QSxWF-7$$\"3m****\\7G$R<)FM$\"3=Cd-EGkuVF-7$$\"3GL LL3x&)*3\"F-$\"3_OPBwhg\"H%F-7$$\"3))**\\i!R(*Rc\"F-$\"3_>=#>.RX<%F-7$ $\"3umm\"H2P\"Q?F-$\"3i(\\k&\\i?zSF-7$$\"3MLL$eRwX5$F-$\"3N$[KdiY0\"RF -7$$\"33ML$3x%3yTF-$\"3#o:P-IX(yPF-7$$\"3emm\"z%4\\Y_F-$\"3q!Rb\")yJ1n $F-7$$\"3`LLeR-/PiF-$\"3;f#4=V^Xe$F-7$$\"3]***\\il'pisF-$\"3a(=z()eZg] $F-7$$\"3>MLe*)>VB$)F-$\"3-\")\\0W'yOV$F-7$$\"3Y++DJbw!Q*F-$\"3S&GfZMb 'oLF-7$$\"3%ommTIOo/\"!#<$\"3,![(*o(y!yI$F-7$$\"3YLL3_>jU6Fhr$\"3In2AU WXeKF-7$$\"37++]i^Z]7Fhr$\"3Y2^+&fIp?$F-7$$\"33++](=h(e8Fhr$\"3!y&p#3 \")**)eJF-7$$\"3/++]P[6j9Fhr$\"3tb$[<**fc6$F-7$$\"3UL$e*[z(yb\"Fhr$\"3 G;mdLdnyIF-7$$\"3wmm;a/cq;Fhr$\"3#)H#G/F/s.$F-7$$\"3%ommmJFhr $\"3JA?@9EXOHF-7$$\"3K+]i!f#=$3#Fhr$\"3_uYGF-7$$\"3GLL3_?`(\\#Fhr$\"3U39yGm,$z#F-7$$\"3fL$e*)>pxg #Fhr$\"3!*y(R5sfjw#F-7$$\"33+]Pf4t.FFhr$\"3A.cTc#4Su#F-7$$\"3uLLe*Gst! GFhr$\"3TOUQcBq?FF-7$$\"30+++DRW9HFhr$\"3'fZ#>RBZ(p#F-7$$\"3:++DJE>>IF hr$\"3K*\\G.HEbn#F-7$$\"3F+]i!RU07$Fhr$\"3?A&[ar!)\\l#F-7$$\"3+++v=S2L KFhr$\"3D)RM_#F-7$$\"3Ymm;/OgbRFhr$\"3!H(e'Gqvp]#F-7$$\"3w** \\ilAFjSFhr$\"3G.Q%\\8\">!\\#F-7$$\"3yLLL$)*pp;%Fhr$\"3ib91[;WuCF-7$$ \"3)RL$3xe,tUFhr$\"3YiZ'G!yteCF-7$$\"3Cn;HdO=yVFhr$\"3#>?#[Z%Fhr$\"3&=#eB\\2\"*HCF-7$$\"3SnmT&G!e&e%Fhr$\"31l&)3`Uk9CF -7$$\"3#RLLL)Qk%o%Fhr$\"3Ji/h4TI,CF-7$$\"37+]iSjE!z%Fhr$\"3Wp*>E'RR(Q# F-7$$\"3a+]P40O\"*[Fhr$\"3Cft+H3PuBF-7$$\"\"&F*$\"31)*y*\\xz1O#F-F.-%+ AXESLABELSG6%Q!6\"F\\_l%(DEFAULTG-%%VIEWG6$;F2Fe^l;F2$\"\"'!\"\"" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " }}}}{PARA 0 "" 0 "" {TEXT -1 99 "\nWe can \"check\" the limit by zoo ming in towards the \"missing point\" so as to plot over an interval \+ " }{XPPEDIT 18 0 "[4-delta, 4+delta];" "6#7$,&\"\"%\"\"\"%&deltaG!\"\" ,&F%F&F'F&" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT -1 34 " is a small positive real number.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "delta := 0.001;\ncurve := plot((sqrt(x)-2)/( x-4),x=4-delta..4+delta,color=blue):\nmissingpoint := plot([[4,0.25]], style=point,symbol=circle,color=blue):\nplots[display]([curve,missingp oint]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&deltaG$\"\"\"!\"$" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6&7#7$ $\"\"%\"\"!$\"3++++++++D!#=-%'COLOURG6&%$RGBG$F*F*F2$\"*++++\"!\")-%&S TYLEG6#%&POINTG-%'SYMBOLG6#%'CIRCLEG-F$6$7S7$$\"35++++++**R!#<$\"3izR` pi:+DF-7$$\"3cL$3VfV!**RFD$\"3X&)oDc%\\,]#F-7$$\"3i;H[D:3**RFD$\"3^v47 GN9+DF-7$$\"3GLe0$=C\"**RFD$\"3H34Qho8+DF-7$$\"3YL3RBr;**RFD$\"3I_:_], 8+DF-7$$\"3'o\"zjf)4#**RFD$\"3DSlirM7+DF-7$$\"3g$e4;[\\#**RFD$\"3ZO@]z s6+DF-7$$\"3=]i'y]!H**RFD$\"3kY*zz'36+DF-7$$\"3g$ezs$HL**RFD$\"3oX1CPU 5+DF-7$$\"3A]7iI_P**RFD$\"3o\"QTyi(4+DF-7$$\"3'pm@Xt=%**RFD$\"3qg0LH34 +DF-7$$\"3_L3y_qX**RFD$\"3@oJET[3+DF-7$$\"3-+]1!>+&**RFD$\"3-OF=+\"y+] #F-7$$\"31+]Z/Na**RFD$\"38-u\\J82+DF-7$$\"33+]$fC&e**RFD$\"3BeSo3[1+DF -7$$\"3m$ez6:B'**RFD$\"3LXIS&))e+]#F-7$$\"3ym;=C#o'**RFD$\"3L***=@%=0+ DF-7$$\"3ymm#pS1(**RFD$\"3v\"y*fve/+DF-7$$\"3O]i`A3v**RFD$\"3c0L>N*Q+] #F-7$$\"3#pmwy8!z**RFD$\"3py(==zK+]#F-7$$\"3C]i.tK$)**RFD$\"3&4&\\j^g- +DF-7$$\"3S](3zMu)**RFD$\"3**)*QWL'>+]#F-7$$\"3%p;H_?<***RFD$\"3'pD>o$ H,+DF-7$$\"3o;zihl&***RFD$\"3-y[G(y1+]#F-7$$\"3eL3#G,*****RFD$\"3XOf2a ,++DF-7$$\"3F$ezw5V++%FD$\"3%zY!\\kK***\\#F-7$$\"3W]PQ#\\\"3+SFD$\"3]0 f$pE()**\\#F-7$$\"3?Le\"*[H7+SFD$\"37NQ^*y!)**\\#F-7$$\"3O++dxd;+SFD$ \"3+*o\"z(4u**\\#F-7$$\"3\"**\\_qn2-+%FD$\"3'yh$H^v'**\\#F-7$$\"3K]i&p @[-+%FD$\"3U**))H<7'**\\#F-7$$\"3A+vgHKH+SFD$\"3iyfb%=a**\\#F-7$$\"3'p mwanL.+%FD$\"3Kz`Qly%**\\#F-7$$\"3r+]2goP+SFD$\"3U&H&Q=6%**\\#F-7$$\"3 =$eR<*fT+SFD$\"3C3On/]$**\\#F-7$$\"3_+])Hxe/+%FD$\"3%z(f$3KG**\\#F-7$$ \"3!o\"H!o-*\\+SFD$\"3G66#>.A**\\#F-7$$\"3P]7k.6a+SFD$\"3-vwFea\"**\\# F-7$$\"3>n;WTAe+SFD$\"3o4[RJ!4**\\#F-7$$\"3B]i!*3`i+SFD$\"3;!G=JI-**\\ #F-7$$\"3`LL*zym1+%FD$\"3G&egI#e*)*\\#F-7$$\"3#Q$3N1#42+%FD$\"3[7BL'>* ))*\\#F-7$$\"3P " 0 "" {MPLTEXT 1 0 189 "delta := 0.0001;\ncurve := plot((sqrt(x)-2)/(x-4),x=4-delta..4+de lta,color=blue):\nmissingpoint := plot([[4,0.25]],style=point,symbol=c ircle,color=blue):\nplots[display]([curve,missingpoint]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&deltaG$\"\"\"!\"%" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6&7#7$$\"\"%\"\"!$\"3++ ++++++D!#=-%'COLOURG6&%$RGBG$F*F*F2$\"*++++\"!\")-%&STYLEG6#%&POINTG-% 'SYMBOLG6#%'CIRCLEG-F$6$7S7$$\"3C+++++!***R!#<$\"3tI+>Dc,+DF-7$$\"3WL3 VfV!***RFD$\"3j,\">S%\\,+DF-7$$\"3!=H[D:3***RFD$\"3_\"3@8N9+]#F-7$$\"3 k$e0$=C\"***RFD$\"3&oe*z%o8+]#F-7$$\"3_$3RBr;***RFD$\"3_J:#Q,8+]#F-7$$ \"3'=zjf)4#***RFD$\"3]*HsgM7+]#F-7$$\"3me4;[\\#***RFD$\"3D^=&ps6+]#F-7 $$\"3)\\i'y]!H***RFD$\"3;Yr!f36+]#F-7$$\"3Oezs$HL***RFD$\"3P7)[HU5+]#F -7$$\"3CD@1Bv$***RFD$\"3)R]*3i(4+]#F-7$$\"3ym@Xt=%***RFD$\"3)\\1ZB34+] #F-7$$\"3O$3y_qX***RFD$\"3)HqBO[3+]#F-7$$\"3/+l+>+&***RFD$\"3Ev1f4y++D F-7$$\"39+vW]V&***RFD$\"3X)o%yKr++DF-7$$\"35+NfC&e***RFD$\"3&4dg0[1+]# F-7$$\"3Sez6:B'***RFD$\"3k;-J))e++DF-7$$\"3em\"=C#o'***RFD$\"3Lf]*R=0+ ]#F-7$$\"3smEpS1(***RFD$\"3,W)yte/+]#F-7$$\"3'[i`A3v***RFD$\"32vXR$*Q+ +DF-7$$\"3umwy8!z***RFD$\"3>V$3\"zK++DF-7$$\"3%\\i.tK$)***RFD$\"3Z++DF-7$$\"3_;H_?<****RFD$\"3Q\") Gt$H,+]#F-7$$\"3%=zihl&****RFD$\"3mLe$)y1++DF-7$$\"3]$3#G,******RFD$\" 3fPG\\A+++DF-7$$\"3fezw5V++SFD$\"3SbNxE$****\\#F-7$$\"3yu$Q#\\\"3++%FD $\"3WhWtE()***\\#F-7$$\"30$e\"*[H7++%FD$\"3AQqyy!)***\\#F-7$$\"3/+qvxl ,+SFD$\"3ugM%)4u***\\#F-7$$\"3s\\_qn2-+SFD$\"3?m?.bn***\\#F-7$$\"37Dcp @[-+SFD$\"3!H(=b@h***\\#F-7$$\"3?]2'HKH++%FD$\"3w59L=a***\\#F-7$$\"3Dm wanL.+SFD$\"3//GR'y%***\\#F-7$$\"3a*\\2goP++%FD$\"3+a*H;6%***\\#F-7$$ \"3'z&R<*fT++%FD$\"33Qc:+N***\\#F-7$$\"3q*\\)Hxe/+SFD$\"3kk-nJG***\\#F -7$$\"3:\"H!o-*\\++%FD$\"3mmtt-A***\\#F-7$$\"3]CTO5T0+SFD$\"3Ue?LX:*** \\#F-7$$\"3XmT9C#e++%FD$\"3/Nxd-4***\\#F-7$$\"3eC1*3`i++%FD$\"3]#*QkH- ***\\#F-7$$\"3`L$*zym1+SFD$\"3S&=)\\\"e*)**\\#F-7$$\"3&G3N1#42+SFD$\"3 C*>_(=*))**\\#F-7$$\"3c\"HYt7v++%FD$\"3')[`[h#))**\\#F-7$$\"3`**p(G**y ++%FD$\"3Oou]dw)**\\#F-7$$\"3>;9@BM3+SFD$\"3'e5V_'p)**\\#F-7$$\"3=L`v& Q(3+SFD$\"3u\\<7Yj)**\\#F-7$$\"3%[i`1h\"4+SFD$\"3)**R$*fo&)**\\#F-7$$ \"3uu.Uac4+SFD$\"3a;')>a])**\\#F-7$$\"3x**********4+SFD$\"3c\\'4_P%)** \\#F-F.-%+AXESLABELSG6%Q!6\"Fd\\l%(DEFAULTG-%%VIEWG6$;$\"&***R!\"%$\"& ,+%F]]lFf\\l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 29 "We can illustrate the limit " }{XPPEDIT 18 0 "limit((1+x)^(1/x),x = 0) = exp(1);" "6#/-%&limitG6 $),&\"\"\"F)%\"xGF)*&F)F)F*!\"\"/F*\"\"!-%$expG6#F)" }{TEXT -1 25 " by drawing the graph of " }{XPPEDIT 18 0 "y = (1+x)^(1/x);" "6#/%\"yG),& \"\"\"F'%\"xGF'*&F'F'F(!\"\"" }{TEXT -1 34 " and putting a \"hole\" at the point" }{XPPEDIT 18 0 "``(0,exp(1));" "6#-%!G6$\"\"!-%$expG6#\"\" \"" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "cur ve := plot((1+x)^(1/x),x=-0.7..2,color=blue):\nmissingpoint := plot([[ 0,exp(1.0)]],style=point,symbol=circle,color=blue):\nplots[display]([c urve,missingpoint],view=[-0.7..2,0..5.6]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7V7$$!3a************* *p!#=$\"3[%=#=/:J%e&!#<7$$!3@++v=%Qdq'F*$\"3=FPBs/zP_F-7$$!3v****\\PoZ 6kF*$\"3/`&=yKv`%\\F-7$$!3#)*\\i!49WbhF*$\"3JL#y,B?bs%F-7$$!3))**\\i!) fS**eF*$\"3/.IY>pmJXF-7$$!3(*)\\PMOn9h&F*$\"3')RD&zc0#RVF-7$$!3<***\\i uGNK&F*$\"3Y'>$yQw5pTF-7$$!3>***\\PAMQu%F*$\"3km4Cu*z*zQF-7$$!3a**\\7) )[*o;%F*$\"33s.y]:%fk$F-7$$!3!)**\\iq#)*>j$F*$\"3512k(H)[kMF-7$$!3!*** \\i0Q9yIF*$\"3*[8UMJLUI$F-7$$!3\\**\\iDnM0DF*$\"3Km#zo#*G<;$F-7$$!3#)* *\\78mQM>F*$\"37)*ou/ZTQIF-7$$!3K****\\dR3Z8F*$\"3\")3^0%*GKFHF-7$$!3= )***\\(euyH)!#>$\"39k$yR+vo(>e8fF*$\"3Uy*H$R`y$>#F-7$$\"3k++D\"3tm['F*$\"3cwkX;)49;#F -7$$\"35,]PuO&>3(F*$\"3e%HO6$e')H@F-7$$\"3%4+D1=Z,g(F*$\"3(f8bH^zR5#F- 7$$\"3!>+]PO5)f\")F*$\"3e+YJgD_x?F-7$$\"3;+++&>(*zt)F*$\"3#Q'HI$4!p^?F -7$$\"3?***\\(3-k.$*F*$\"3J6W)>Mqx-#F-7$$\"3w+]P4*G4&)*F*$\"3#*zW\\#Q( z0?F-7$$\"3)***\\7q*fe/\"F-$\"3SDl8[>k#)>F-7$$\"3!*****\\$*=Y+6F-$\"3O ()fO*>lG'>F-7$$\"35++D,6we6F-$\"3&[pJaNVF%>F-7$$\"3#**\\P%[))e67F-$\"3 X@\"[,*RLD>F-7$$\"3?++vHNMp7F-$\"39\")3%**eMr!>F-7$$\"3/+v$R='oB8F-$\" 3iAd6$R_2*=F-7$$\"37+vo:**[!Q\"F-$\"3'[>2g*fMu=F-7$$\"3z***\\i%f-O9F-$ \"3)3z!=b$p*e=F-7$$\"3=+vVBq;%\\\"F-$\"3I[jz!HHN%=F-7$$\"3M+++\"zj,b\" F-$\"3--m]w]DH=F-7$$\"3D+]it&Gug\"F-$\"3ky6t.bA:=F-7$$\"3Q+v$\\<>Um\"F -$\"3D'**))=DX=!=F-7$$\"33++]RQS;F-$\"3tO%=?.=Ku\"F-7$$\"\"#\"\"!$\"3 ?x)ov!30K " 0 "" {MPLTEXT 1 0 183 "delta := 0.001;\ncurve := plot((1+x)^(1/x),x=-delta..delta,color=blue):\nmissingpoint := plot([ [0,exp(1.0)]],style=point,symbol=circle,color=blue):\nplots[display]([ curve,missingpoint]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&deltaG$\" \"\"!\"$" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'C URVESG6&7#7$$\"\"!F)$\"3%)*****z#=G=F!#<-%'COLOURG6&%$RGBGF(F($\"*++++ \"!\")-%&STYLEG6#%&POINTG-%'SYMBOLG6#%'CIRCLEG-F$6$7S7$$!3-+++++++5!#? $\"3!44Vk@U'>FF,7$$!3mmmm;p0k&*!#@$\"3EarAfGe>FF,7$$!3iLL$3FF,7$$!3!pmm;Wp\"e()FH$\"3pc(eN9t%>FF,7$$!3Nnmm\"4m(G$)FH$\"3S +(o+p9%>FF,7$$!33LL$3i.9!zFH$\"3\"p@q*=lN>FF,7$$!3,nm;/R=0vFH$\"3L\"4* )4f-$>FF,7$$!3_++]P8#\\4(FH$\"3Ig7#evY#>FF,7$$!3/nm;/siqmFH$\"3\"\\+Ia ,*=>FF,7$$!3o++](y$pZiFH$\"3!44'ok98>FF,7$$!34LLL$yaE\"eFH$\"3+N1LrA2> FF,7$$!3*pmm;>s%HaFH$\"3F@VqP,->FF,7$$!3!*******\\$*4)*\\FH$\"3h@A(>Xh *=FF,7$$!3%3+++Db\\c%FH$\"3u'RW*HD!*=FF,7$$!3#*******\\1aZTFH$\"3N%o!3 _d%)=FF,7$$!3Enm;/#)[oPFH$\"375q5'>%z=FF,7$$!3iLLL$=exJ$FH$\"3W#zJd*Gt =FF,7$$!3Jd?' =FF,7$$!3gMLLL7i)4#FH$\"3bf)Qb6n&=FF,7$$!39++]P'psm\"FH$\"3$*)HN&o%3&= FF,7$$!3S++]74_c7FH$\"30M-.FEX=FF,7$$!3yKLL$3x%z#)!#A$\"3%3$)enO%R=FF, 7$$!3_NLL3s$QM%F\\t$\"3>!fv'p3M=FF,7$$!3%H#ymm;zr)*!#C$\"3DcG9qJG=FF,7 $$\"3'**zr#F,7$$\"3k)***\\i&p@[#FH $\"3`!f\"QVX%zr#F,7$$\"3W)****\\2'HKHFH$\"3(zR(>&R$)yr#F,7$$\"3*\\mmmw anL$FH$\"3$phQ^XGyr#F,7$$\"3[+++]2goPFH$\"3brGR*zpxr#F,7$$\"3mKL$eR<*f TFH$\"3y'4zDl;xr#F,7$$\"3c+++])Hxe%FH$\"3&R*QV`&ewr#F,7$$\"3'Qmm\"H!o- *\\FH$\"3HgS#3*Qg " 0 "" {MPLTEXT 1 0 185 "delta := 0.00001;\ncurve := plot((1+x)^(1/x),x=-delta..delta,color=blue):\nmissingpoint := plot([ [0,exp(1.0)]],style=point,symbol=circle,color=blue):\nplots[display]([ curve,missingpoint]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&deltaG$\" \"\"!\"&" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'C URVESG6$7S7$$!33+++++++5!#A$\"3iGq)*>aH=F!#<7$$!33nmm;p0k&*!#B$\"3OL9 \\F[H=FF-7$$!3cLL$3J%H=FF-7$$!3InmmT%p\"e()F1$\"3KIY8 KPH=FF-7$$!39nmm\"4m(G$)F1$\"33A&4&[JH=FF-7$$!3;ML$3i.9!zF1$\"3(H,Qwc# H=FF-7$$!33nm;/R=0vF1$\"3mo(e\"H?H=FF-7$$!3U++]P8#\\4(F1$\"3buD\\r9H=F 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*pztPp#=FF-7$$\"3U.+]P?Wl&*F1$\"3YpM!y#)o#=FF-7$$\"33+++++++5F*$\"3^#o (>P#o#=FF--%'COLOURG6&%$RGBG$\"\"!F`[lF_[l$\"*++++\"!\")-F$6&7#7$F_[l$ \"3%)*****z#=G=FF-F[[l-%'SYMBOLG6#%'CIRCLEG-%&STYLEG6#%&POINTG-%+AXESL ABELSG6%Q\"x6\"Q!Ff\\l-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!\"\"!\"&$\"\"\" Fb]lF[]l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur ve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{PARA 0 "" 0 "" {TEXT -1 63 "Find the following limits \+ by any (or all) of the above methods." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit((x^2-x-6)/(x-3),x = 3);" "6#-%&LimitG6$*&,(*$%\"xG\"\"#\"\"\"F)!\"\"\"\"'F,F+,&F)F+\"\" $F,F,/F)F/" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "Limit((x^2-x-6)/(x-3),x=3) ;\n``=Limit(``(factor(numer(op(1,%))))/``(factor(denom(op(1,%)))),op(2 ,%));\n``=eval(subs(``=(_X->_X),rhs(%)));\n``=value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&,(*$)%\"xG\"\"#\"\"\"F,F*!\"\" \"\"'F-F,,&F*F,\"\"$F-F-/F*F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G- %&LimitG6$*&-F$6#*&,&%\"xG\"\"\"\"\"#F.F.,&F-F.\"\"$!\"\"F.F.-F$6#F0F2 /F-F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,&%\"xG\"\"\" \"\"#F*/F)\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G\"\"&" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "f := x -> (x^2-x-6)/(x-3):\n'f(x)'=f(x);\nplot([f(x),[[3,5]]$2],x =-1..5,y=0..f(5),style=[line,point$2],\n symbol=circle,symbolsiz e=[15,18],color=[red,brown$2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"fG6#%\"xG*&,(*$)F'\"\"#\"\"\"F-F'!\"\"\"\"'F.F-,&F'F-\"\"$F.F." }} {PARA 13 "" 1 "" {GLPLOT2D 297 304 304 {PLOTDATA 2 "6(-%'CURVESG6&7S7$ $!\"\"\"\"!$\"\"\"F*7$$!3/+++]2<#p)!#=$\"36+++DHyI6!#<7$$!3[++]7bBavF0 $\"3&****\\([kdW7F37$$!3++++D$3XF'F0$\"35++]n\"\\DP\"F37$$!3c*****\\F) H')\\F0$\"3;++]s,P,:F37$$!3J++]i3@/PF0$\"3')***\\P\"*y&H;F37$$!3V++]7< b:DF0$\"3')***\\(G[W[F37$$\"3m****\\P'=pD\"F0$\"3>++vj=pD@F37$$ \"3y+++]c.iDF0$\"3')*****\\c.iD#F37$$\"3;+++DMe6PF0$\"3C++]U$e6P#F37$$ \"32,++]>q0]F0$\"3))*****\\>q0]#F37$$\"3h******\\U80jF0$\"3=+++DM^IEF3 7$$\"3'4+++0ytb(F0$\"3))*****\\!ytbFF37$$\"3w****\\(QNXp)F0$\"3?++vQNX pGF37$$\"3.+++XDn/5F3$\"3e*****\\asY+$F37$$\"3.+++!y?#>6F3$\"3.+++!y?# >JF37$$\"3'****\\(3wY_7F3$\"3'****\\(3wY_KF37$$\"3#)******HOTq8F3$\"3# )******HOTqLF37$$\"37++v3\">)*\\\"F3$\"3o***\\(3\">)*\\$F37$$\"3:++DEP /B;F3$\"3:++DEP/BOF37$$\"3=++](o:;v\"F3$\"3k++](o:;v$F37$$\"3=++v$)[op =F3$\"3u***\\P)[opQF37$$\"3%*****\\i%Qq*>F3$\"3]****\\i%Qq*RF37$$\"3&* ***\\(QIKH@F3$\"3&****\\(QIKHTF37$$\"3#****\\7:xWC#F3$\"3#****\\7:xWC% 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"6#-%&LimitG6$*&,&*$%\"xG\"\"#\"\"\"\"#D!\"\"F+,&F)F+\"\"&F-F-/F )F/" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "Limit((x^2-25)/(x-5),x=5);\n``=Lim it(``(factor(numer(op(1,%))))/``(factor(denom(op(1,%)))),op(2,%));\n`` =eval(subs(``=(_X->_X),rhs(%)));\n``=value(rhs(%));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%&LimitG6$*&,&*$)%\"xG\"\"#\"\"\"F,\"#D!\"\"F,,&F*F, \"\"&F.F./F*F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*&-F$ 6#*&,&%\"xG\"\"\"\"\"&!\"\"F.,&F-F.F/F.F.F.-F$6#F,F0/F-F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,&%\"xG\"\"\"\"\"&F*/F)F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "f := x -> (x^2-25)/( x-5):\n'f(x)'=f(x);\nplot([f(x),[[5,10]]$2],x=-1..10,y=0..f(10),style= [line,point$2],\n symbol=circle,symbolsize=[15,18],color=[red,br own$2]);" }}{PARA 11 "" 1 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{TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 47 "___________ ____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 47 "_______________________________ ________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit((x^3-27)/(x-3),x = 3);" "6#-%&LimitG6$*&,&*$ %\"xG\"\"$\"\"\"\"#F!\"\"F+,&F)F+F*F-F-/F)F*" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "An s " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 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"_______________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q16" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limi t((sin*x-x)/(x^3),x = 0);" "6#-%&LimitG6$*&,&*&%$sinG\"\"\"%\"xGF*F*F+ !\"\"F**$F+\"\"$F,/F+\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 47 "___________ ____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 47 "_______________________________ ________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 28 "Code for pictures and tables" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 19 "Table calculations " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "f := x -> (x^3-8)/(x-2);\nh := 0.2;\npts:=[seq([h*i,evalf(f(h*i), 4)],i=0..15)]:\nA := array(1..2,1..16):\nfor j to 16 do\n A[1,j] := \+ pts[j,1]:\n A[2,j] := pts[j,2];\nend do:\nevalm(A)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 283 "f := x - > (x^3-8)/(x-2);\nh := 0.2;\npts:=[seq([h*i,evalf(f(h*i),4)],i=0..15)] :\nA := array(1..8,1..2):\nfor j to 8 do\n A[j,1] := pts[j,1]:\n A [j,2] := pts[j,2];\nend do:\nevalm(A);\nB := array(1..8,1..2):\nfor j \+ to 8 do\n B[j,1] := pts[j+8,1]:\n B[j,2] := pts[j+8,2];\nend do:\n evalm(B);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 293 "f := x -> (x^3-8)/(x-2);\nh := 0.05;\npts2:=[seq([ 2+h*i,evalf(f(2+h*i),6)],i=-7..6)]:\nA := array(1..7,1..2):\nfor j to \+ 7 do\n A[j,1] := pts2[j,1]:\n A[j,2] := pts2[j,2];\nend do:\nevalm (A);\nB := array(1..7,1..2):\nfor j to 7 do\n B[j,1] := pts2[j+7,1]: \n B[j,2] := pts2[j+7,2];\nend do:\nevalm(B);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 37 "limit of f(x)=x^2 as x approaches 2 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1076 "f := x -> x^2:\na := 1.7: b := 2.3: c := 2:\np1 := plot(f(x),x =0..2.5,color=red):\np2 := plot([[[a,0],[a,f(a)],[0,f(a)]],\n [[ b,0],[b,f(b)],[0,f(b)]]],color=COLOR(RGB,.4,.4,.4),linestyle=2):\npts \+ := [[a,0],[a,f(a)],[0,f(a)],[b,0],[b,f(b)],[0,f(b)]]:\np3 := plot([pts $3],style=point,symbol=[circle,diamond,cross],color=black):\np4 := plo t([[[c,0],[c,f(c)],[0,f(c)]]$3],style=point,\n symbol= [circle,diamond,cross],color=blue,symbolsize=[15$3]):\np5 := plot([[c, 0],[c,f(c)],[0,f(c)]],color=blue):\np6 := plottools[arrow]([a,-.2],[c- .05,-.2],0,.13,.13,arrow,color=black):\np7 := plottools[arrow]([b,-.2] ,[c+.05,-.2],0,.13,.13,arrow,color=black):\np8 := plottools[arrow]([-. 1,f(a)],[-.1,f(c)-.1],0,.05,.1,arrow,color=black):\np9 := plottools[ar row]([-.1,f(b)],[-.1,f(c)+.1],0,.05,.1,arrow,color=black):\nt1 := plot s[textplot]([[2.5,-.2,`x`],[-.1,6.2,`y`]]):\nt2 := plots[textplot]([[2 ,-.2,`2`],[.2,4.3,`f(2) = 4`],\n [1.95,4.35,`(2,4)`]],color=bl ue):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,p8,p9,t1,t2],\n tic kmarks=[0,0],axes=normal,view=[-.2..2.5,-.3..6.25]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1235 "f := x -> x^2:\nc := 2:\np1 := plot(f(x),x=0 ..2.5,color=red):\np2 := plot([[[c,0],[c,f(c)],[0,f(c)]]$3],style=poin t,\n symbol=[circle,diamond,cross],color=black):\np3 := plot([[c,0], [c,f(c)],[0,f(c)]],color=black,linestyle=2):\nnumframes := 30:\nh := . 3/(numframes-1):\nfrms := NULL:\nfor i from 0 to numframes-1 do \n a := c-h*(numframes-1-i)+.001; \n b := c+h*(numframes-1-i)-.001; \n \+ p4 := plot([[a,0],[a,f(a)],[0,f(a)]],\n color=COLOR(RGB,0,.7,0), linestyle=2):\n p5 := plot([[b,0],[b,f(b)],[0,f(b)]],\n color=C OLOR(RGB,.5,0,1),linestyle=2):\n p6 := plot([[[a,0],[a,f(a)],[0,f(a) ]]$3],\n style=point,symbol=[circle,diamond,cross],\n \+ color=COLOR(RGB,0,.7,0));\n p7 := plot([[[b,0],[b,f(b)],[0,f(b)]]$3 ],\n style=point,symbol=[circle,diamond,cross],\n co lor=COLOR(RGB,.5,0,1));\n t1 := plots[textplot]([[-.2,f(a)-.15,evalf (f(a),3)],\n [a-.15,-.2,evalf(a,3)]],color=COLOR(RGB,0,.7,0)): \n t2 := plots[textplot]([[-.2,f(b)+.15,evalf(f(b),3)],\n [b+ .15,-.2,evalf(b,3)]],color=COLOR(RGB,.5,0,1)): \n frms := frms,plot s[display]([p1,p2,p3,p4,p5,p6,p7,t1,t2],\n view=[-.2..2.8,-.5 ..6.5]);\nend do: \nplots[display]([frms],insequence=true,labels=[`x`, `y`],\n tickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 37 "limit of f(x)=1/x as x approaches 3 " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1085 "f := x - > 1/x:\na := 2.3: b := 3.7: c := 3:\np1 := plot(f(x),x=0.9..4.5,color= red):\np2 := plot([[[a,0],[a,f(a)],[0,f(a)]],\n [[b,0],[b,f(b)], [0,f(b)]]],color=COLOR(RGB,.4,.4,.4),linestyle=2):\npts := [[a,0],[a,f (a)],[0,f(a)],[b,0],[b,f(b)],[0,f(b)]]:\np3 := plot([pts$3],style=poin t,symbol=[circle,diamond,cross],color=black):\np4 := plot([[[c,0],[c,f (c)],[0,f(c)]]$3],style=point,\n symbol=[circle,diamon d,cross],color=blue,symbolsize=[15$3]):\np5 := plot([[c,0],[c,f(c)],[0 ,f(c)]],color=blue):\np6 := plottools[arrow]([a,-.05],[c-.07,-.05],0,. 03,.1,arrow,color=black):\np7 := plottools[arrow]([b,-.05],[c+.07,-.05 ],0,.03,.1,arrow,color=black):\np8 := plottools[arrow]([-.1,f(a)],[-.1 ,f(c)+.02],0,.1,.4,arrow,color=black):\np9 := plottools[arrow]([-.1,f( b)],[-.1,f(c)-.02],0,.1,.4,arrow,color=black):\nt1 := plots[textplot]( [[4.5,-.03,`x`],[-.1,1.1,`y`]]):\nt2 := plots[textplot]([[3,-.05,`3`], [.35,.38,`f(3) = 1/3`],\n [3.1,.4,`(3,1/3)`]],color=blue):\npl ots[display]([p1,p2,p3,p4,p5,p6,p7,p8,p9,t1,t2],\n tickmarks=[ 0,0],axes=normal,view=[-.15..4.5,-.07..1.1]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1238 "f := x -> \+ 1/x:\nc := 3:\np1 := plot(f(x),x=0.9..4.5,color=red):\np2 := plot([[[c ,0],[c,f(c)],[0,f(c)]]$4],style=point,\n symbol=[circle,diamond,cros s],color=black):\np3 := plot([[c,0],[c,f(c)],[0,f(c)]],color=black,lin estyle=2):\nnumframes := 20:\nh := 1./(numframes-1):\nfrms := NULL:\nf or i from 0 to numframes-1 do \n a := c-h*(numframes-1-i)+.001; \n \+ b := c+h*(numframes-1-i)-.001; \n p4 := plot([[a,0],[a,f(a)],[0,f(a )]],\n color=COLOR(RGB,0,.7,0),linestyle=2):\n p5 := plot([[b,0 ],[b,f(b)],[0,f(b)]],\n color=COLOR(RGB,.5,0,1),linestyle=2):\n \+ p6 := plot([[[a,0],[a,f(a)],[0,f(a)]]$3],\n style=point,symbol =[circle,diamond,cross],\n color=COLOR(RGB,0,.7,0));\n p7 \+ := plot([[[b,0],[b,f(b)],[0,f(b)]]$3],\n style=point,symbol=[ci rcle,diamond,cross],\n color=COLOR(RGB,.5,0,1));\n t1 := \+ plots[textplot]([[-.4,f(a)+.03,evalf(f(a),3)],\n [a-.2,-.05,eva lf(a,3)]],color=COLOR(RGB,0,.7,0)):\n t2 := plots[textplot]([[-.4,f( b)-.03,evalf(f(b),3)],\n [b+.2,-.05,evalf(b,3)]],color=COLOR(RG B,.5,0,1)): \n frms := frms,plots[display]([p1,p2,p3,p4,p5,p6,p7,t1 ,t2],\n view=[-.4..4.5,-.05..1.1]);\nend do: \nplots[display] ([frms],insequence=true,labels=[`x`,`y`],\n tickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }