{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Lab 6 Maple Activities for September 19: (Lesson 3 - One-Sided Limits)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "1. If f(" }{XPPEDIT 18 0 "x;" "6# %\"xG" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "PIECEWISE([x^2, x <= 0],[2*x -1, 0 < x]);" "6#-%*PIECEWISEG6$7$*$%\"xG\"\"#1F(\"\"!7$,&*&F)\"\"\"F( F/F/F/!\"\"2F+F(" }{TEXT -1 9 " , then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 " (a) use the graph of f to \+ find " }{XPPEDIT 18 0 "Limit(f(x),x = 0,right)" "6#-%&LimitG6%-%\"fG6 #%\"xG/F)\"\"!%&rightG" }{TEXT -1 5 " , " }{XPPEDIT 18 0 "Limit(f(x) ,x = 0,left)" "6#-%&LimitG6%-%\"fG6#%\"xG/F)\"\"!%%leftG" }{TEXT -1 11 " , and " }{XPPEDIT 18 0 "Limit(f(x),x = 0)" "6#-%&LimitG6$-%\" fG6#%\"xG/F)\"\"!" }{TEXT -1 2 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "f := x -> piecewise(x<=0, x^2, x>0, 2*x - 1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 55 "plot(f(x), x=-3..3, y=-2..10, discont=true, co lor=red);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 " (b) find " }{XPPEDIT 18 0 "Limit(f(x),x = 0,right) " "6#-%&LimitG6%-%\"fG6#%\"xG/F)\"\"!%&rightG" }{TEXT -1 13 " by lett ing " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 62 " approach 0 from valu es that are larger than 0. That is, let " }{XPPEDIT 18 0 "x" "6#%\"xG " }{TEXT -1 28 " approach 0 from the right. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "f(0.1); f(0.001); f(0.00001); f(0.0000001); f(0. 0000000001);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 21 " It appears that " }{XPPEDIT 18 0 "Limit(f(x),x = 0 ,right)" "6#-%&LimitG6%-%\"fG6#%\"xG/F)\"\"!%&rightG" }{TEXT -1 40 " \+ = - 1. The graph also shows that as " }{XPPEDIT 18 0 "x" "6#%\"xG" } {TEXT -1 32 " approaches 0 from the right, f(" }{XPPEDIT 18 0 "x" "6#% \"xG" }{TEXT -1 18 ") approaches - 1. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 29 " (c) use Maple to f ind " }{XPPEDIT 18 0 "Limit(f(x),x = 0,right)" "6#-%&LimitG6%-%\"fG6# %\"xG/F)\"\"!%&rightG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Limit(f(x), x=0, right); value(%); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 16 " (d) find " }{XPPEDIT 18 0 "Limit(f(x),x = 0,left)" "6#-%&Li mitG6%-%\"fG6#%\"xG/F)\"\"!%%leftG" }{TEXT -1 13 " by letting " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 63 " approach 0 from values that are smaller than 0. That is, let " }{XPPEDIT 18 0 "x" "6#%\"xG" } {TEXT -1 26 " approach 0 from the left." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "f(-0.1); f(-0.001); f(-0.00001); f(-0.0000001); f(-0. 0000000001);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 20 " It appears that " }{XPPEDIT 18 0 "Limit(f(x),x = 0 ,left)" "6#-%&LimitG6%-%\"fG6#%\"xG/F)\"\"!%%leftG" }{TEXT -1 38 " = \+ 0. The graph also shows that as " }{XPPEDIT 18 0 "x" "6#%\"xG" } {TEXT -1 31 " approaches 0 from the left, f(" }{XPPEDIT 18 0 "x" "6#% \"xG" }{TEXT -1 15 ") approaches 0." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 " (e) use Maple to find " }{XPPEDIT 18 0 "Limit(f(x),x = 0,left)" "6#-%&LimitG6%-%\"fG6#%\"xG/F) \"\"!%%leftG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Limit(f(x), x=0, left); value(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 15 " (f) find " }{XPPEDIT 18 0 "Limit(f( x),x = 0)" "6#-%&LimitG6$-%\"fG6#%\"xG/F)\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "Li mit(f(x),x = 0)" "6#-%&LimitG6$-%\"fG6#%\"xG/F)\"\"!" }{TEXT -1 71 " \+ does not exist since the righthand and lefthand limits are not equal \+ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 29 " (g) use Maple to find " }{XPPEDIT 18 0 "Limit(f(x),x = 0 )" "6#-%&LimitG6$-%\"fG6#%\"xG/F)\"\"!" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Limit(f(x),x=0); value(%);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 256 83 "Remember, the correct answer for this limit is does not \+ exist (DNE), not undefined." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 13 "Digits := 15:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "2. If f(" }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 6 ") = " }{XPPEDIT 18 0 "PIECEWISE([-x^2+1, x < 0],[sqrt(x), -x <= 0 and x- 4 <= 0],[x^2-4*x+2, 4 < x]);" "6#-%*PIECEWISEG6%7$,&*$%\"xG\"\"#!\"\" \"\"\"F,2F)\"\"!7$-%%sqrtG6#F)31,$F)F+F.1,&F)F,\"\"%F+F.7$,(*$F)F*F,*& F8F,F)F,F+F*F,2F8F)" }{TEXT -1 9 " , then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 " (a) use the graph of f \+ to find " }{XPPEDIT 18 0 "Limit(f(x),x = 4,right)" "6#-%&LimitG6%-%\"f G6#%\"xG/F)\"\"%%&rightG" }{TEXT -1 5 " , " }{XPPEDIT 18 0 "Limit(f( x),x = 4,left)" "6#-%&LimitG6%-%\"fG6#%\"xG/F)\"\"%%%leftG" }{TEXT -1 12 " , and " }{XPPEDIT 18 0 "Limit(f(x),x = 4)" "6#-%&LimitG6$-% \"fG6#%\"xG/F)\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "f := x -> piecewise(x<0, -x^2 + 1, 0<=x and x<=4, sqrt(x), x>4, x^2 -4*x + 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "plot(f(x), x=-3..7, y=-6..12, disco nt=true, color=red);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 " (b) find " }{XPPEDIT 18 0 "Limit(f(x),x \+ = 4,right);" "6#-%&LimitG6%-%\"fG6#%\"xG/F)\"\"%%&rightG" }{TEXT -1 13 " by letting " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 62 " approac h 4 from values that are larger than 4. That is, let " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 28 " approach 4 from the right. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "f(4.1); f(4.001); f(4.00001); f(4.0 000001); f(4.0000000001);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 21 " It appears that " }{XPPEDIT 18 0 "Li mit(f(x),x = 4,right);" "6#-%&LimitG6%-%\"fG6#%\"xG/F)\"\"%%&rightG" } {TEXT -1 38 " = 2. The graph also shows that as " }{XPPEDIT 18 0 "x " "6#%\"xG" }{TEXT -1 32 " approaches 4 from the right, f(" }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 16 ") approaches 2. " }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 28 " (c) use Map le to find " }{XPPEDIT 18 0 "Limit(f(x),x = 4,right);" "6#-%&LimitG6%- %\"fG6#%\"xG/F)\"\"%%&rightG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Limit(f(x), x=4, \+ right); value(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 16 " (d) find " }{XPPEDIT 18 0 "Limit(f(x),x = \+ 4,left);" "6#-%&LimitG6%-%\"fG6#%\"xG/F)\"\"%%%leftG" }{TEXT -1 13 " \+ by letting " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 63 " approach 4 fr om values that are smaller than 4. That is, let " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 26 " approach 4 from the left." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "f(3.9); f(3.999); f(3.99999); f(3.9999999); f(3.9999999999);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 20 " It appears that " }{XPPEDIT 18 0 "Limit(f(x), x = 4,left);" "6#-%&LimitG6%-%\"fG6#%\"xG/F)\"\"%%%leftG" }{TEXT -1 38 " = 2. The graph also shows that as " }{XPPEDIT 18 0 "x" "6#%\"x G" }{TEXT -1 31 " approaches 4 from the left, f(" }{XPPEDIT 18 0 "x" " 6#%\"xG" }{TEXT -1 15 ") approaches 2." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 " (e) use Maple to fi nd " }{XPPEDIT 18 0 "Limit(f(x),x = 4,left);" "6#-%&LimitG6%-%\"fG6#% \"xG/F)\"\"%%%leftG" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Limit(f(x), x=4, left); value(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 " (f) find " }{XPPEDIT 18 0 "Limit(f(x),x = 4);" "6#-%&LimitG6$-%\"fG6#%\"xG/F)\"\" %" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 " " }{XPPEDIT 18 0 "Limit(f(x),x = 4);" "6#-%&LimitG6 $-%\"fG6#%\"xG/F)\"\"%" }{TEXT -1 14 " = 2 since " }{XPPEDIT 18 0 " Limit(f(x),x = 4,right)" "6#-%&LimitG6%-%\"fG6#%\"xG/F)\"\"%%&rightG" }{TEXT -1 6 " = " }{XPPEDIT 18 0 "Limit(f(x),x = 4,left)" "6#-%&Lim itG6%-%\"fG6#%\"xG/F)\"\"%%%leftG" }{TEXT -1 6 " = 2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 29 " ( g) use Maple to find " }{XPPEDIT 18 0 "Limit(f(x),x = 4)" "6#-%&Limi tG6$-%\"fG6#%\"xG/F)\"\"%" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "Limit(f(x),x=4); value(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Digits := 25:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "3. If f(" }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 6 ") = " } {XPPEDIT 18 0 "PIECEWISE([-x^2+1, x < 0],[sqrt(x), -x <= 0 and x-4 <= \+ 0],[x^2-4*x+2, 4 < x]);" "6#-%*PIECEWISEG6%7$,&*$%\"xG\"\"#!\"\"\"\"\" F,2F)\"\"!7$-%%sqrtG6#F)31,$F)F+F.1,&F)F,\"\"%F+F.7$,(*$F)F*F,*&F8F,F) F,F+F*F,2F8F)" }{TEXT -1 10 " , then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 " (a) use the graph of f to fi nd " }{XPPEDIT 18 0 "Limit(f(x),x = 0,right)" "6#-%&LimitG6%-%\"fG6#% \"xG/F)\"\"!%&rightG" }{TEXT -1 5 " , " }{XPPEDIT 18 0 "Limit(f(x),x = 0,left)" "6#-%&LimitG6%-%\"fG6#%\"xG/F)\"\"!%%leftG" }{TEXT -1 11 " , and " }{XPPEDIT 18 0 "Limit(f(x),x = 0)" "6#-%&LimitG6$-%\"fG6# %\"xG/F)\"\"!" }{TEXT -1 2 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "f := x -> piecewise(x<0, -x^2 + 1, 0<=x and x<=4, sqrt(x), x>4 , x^2 -4*x + 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "plot(f(x), x=-3..7, y=-6.. 12, discont=true, color=red);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 16 " (b) find " }{XPPEDIT 18 0 "Lim it(f(x),x = 0,right)" "6#-%&LimitG6%-%\"fG6#%\"xG/F)\"\"!%&rightG" } {TEXT -1 13 " by letting " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 62 " approach 0 from values that are larger than 0. That is, let " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 28 " approach 0 from the right. \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "f(0.1); f(0.001); f(0.0 0001); f(0.0000001); f(0.0000000001); f(0.00000000000001);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 " It a ppears that " }{XPPEDIT 18 0 "Limit(f(x),x = 0,right)" "6#-%&LimitG6%- %\"fG6#%\"xG/F)\"\"!%&rightG" }{TEXT -1 38 " = 0. The graph also sh ows that as " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 32 " approaches 0 from the right, f(" }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 16 ") appr oaches 0. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 29 " (c) use Maple to find " }{XPPEDIT 18 0 "Limit(f( x),x = 0,right)" "6#-%&LimitG6%-%\"fG6#%\"xG/F)\"\"!%&rightG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "L imit(f(x), x=0, right); value(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 16 " (d) find " }{XPPEDIT 18 0 "Limit(f(x),x = 0,left)" "6#-%&LimitG6%-%\"fG6#%\"xG/F)\"\"!%%leftG" } {TEXT -1 13 " by letting " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 63 " approach 0 from values that are smaller than 0. That is, let " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 26 " approach 0 from the left." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "f(-0.1); f(-0.001); f(-0. 00001); f(-0.0000001); f(-0.0000000001);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 20 " It appears that " } {XPPEDIT 18 0 "Limit(f(x),x = 0,left)" "6#-%&LimitG6%-%\"fG6#%\"xG/F) \"\"!%%leftG" }{TEXT -1 38 " = 1. The graph also shows that as " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 31 " approaches 0 from the left, f(" }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 15 ") approaches 1." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 " \+ (e) use Maple to find " }{XPPEDIT 18 0 "Limit(f(x),x = 0,left)" " 6#-%&LimitG6%-%\"fG6#%\"xG/F)\"\"!%%leftG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 33 "Limit(f(x), x=0, left); value(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 " (f) find \+ " }{XPPEDIT 18 0 "Limit(f(x),x = 0)" "6#-%&LimitG6$-%\"fG6#%\"xG/F)\" \"!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " \+ " }{XPPEDIT 18 0 "Limit(f(x),x = 0)" "6#-%&LimitG6$-%\"fG6#%\"xG/F) \"\"!" }{TEXT -1 71 " does not exist since the righthand and lefthand limits are not equal " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 " " }} {PARA 0 "" 0 "" {TEXT -1 29 " (g) use Maple to find " }{XPPEDIT 18 0 "Limit(f(x),x = 0)" "6#-%&LimitG6$-%\"fG6#%\"xG/F)\"\"!" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Limit(f(x),x=0); value(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 80 "Again, the correct answer for this limit \+ is does not exist (DNE), not undefined." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "4. " } {TEXT 258 48 "Maple has at least one error in its programming:" } {TEXT -1 78 " Ask Maple to find the following three limits, which we \+ did in lecture: (a) " }{XPPEDIT 18 0 "Limit(sqrt(x-4),x = 4,right);" "6#-%&LimitG6%-%%sqrtG6#,&%\"xG\"\"\"\"\"%!\"\"/F*F,%&rightG" }{TEXT -1 10 " , (b) " }{XPPEDIT 18 0 "Limit(sqrt(x-4),x = 4,left);" "6#-% &LimitG6%-%%sqrtG6#,&%\"xG\"\"\"\"\"%!\"\"/F*F,%%leftG" }{TEXT -1 15 " , and (c) " }{XPPEDIT 18 0 "Limit(sqrt(x-4),x = 4);" "6#-%&LimitG 6$-%%sqrtG6#,&%\"xG\"\"\"\"\"%!\"\"/F*F," }{TEXT -1 46 " . Recall ou r answers to these three limits:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 17 "Answer to (a): 0" }}{PARA 0 "" 0 "" {TEXT -1 19 "Answer to (b): DNE" }}{PARA 0 "" 0 "" {TEXT -1 19 "Answe r to (c): DNE" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "Now, look at the answers which Maple gives for these thre e limits:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "Limit(sqrt(x-4),x = 4,right);" "6#- %&LimitG6%-%%sqrtG6#,&%\"xG\"\"\"\"\"%!\"\"/F*F,%&rightG" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Limit(sqrt(x-4),x=4, r ight); value(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }{TEXT 259 5 "NOTE:" }{TEXT -1 25 " This answer \+ is correct." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 5 "(b) " }{XPPEDIT 18 0 "Limit(sqrt(x-4),x = 4,left);" "6# -%&LimitG6%-%%sqrtG6#,&%\"xG\"\"\"\"\"%!\"\"/F*F,%%leftG" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Limit(sqrt(x-4),x=4, l eft); value(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 5 "NOTE:" }{TEXT -1 17 " This answer is " }{TEXT 261 5 "WRONG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "(c) " }{XPPEDIT 18 0 "Lim it(sqrt(x-4),x = 4);" "6#-%&LimitG6$-%%sqrtG6#,&%\"xG\"\"\"\"\"%!\"\"/ F*F," }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Limi t(sqrt(x-4),x=4); value(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 5 "NOTE:" }{TEXT -1 22 " Th is answer is also " }{TEXT 263 5 "WRONG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Problems \+ to practice:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "1. If f(" }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 6 ") = \+ " }{XPPEDIT 18 0 "PIECEWISE([x^2, x <= 2],[3*x-2, 2 < x]);" "6#-%*PIEC EWISEG6$7$*$%\"xG\"\"#1F(F)7$,&*&\"\"$\"\"\"F(F/F/F)!\"\"2F)F(" } {TEXT -1 16 " , then find " }{XPPEDIT 18 0 "Limit(f(x),x = 2);" "6# -%&LimitG6$-%\"fG6#%\"xG/F)\"\"#" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "2. If f(" }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 6 ") = " }{XPPEDIT 18 0 "PIECEWISE([3*x^2- x+1, x <= -1],[2*x^2-4, -1 < x]);" "6#-%*PIECEWISEG6$7$,(*&\"\"$\"\"\" *$%\"xG\"\"#F*F*F,!\"\"F*F*1F,,$F*F.7$,&*&F-F**$F,F-F*F*\"\"%F.2,$F*F. F," }{TEXT -1 16 " , then find " }{XPPEDIT 18 0 "Limit(f(x),x = -1) ;" "6#-%&LimitG6$-%\"fG6#%\"xG/F),$\"\"\"!\"\"" }{TEXT -1 2 " ." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "3. If f(" }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 6 ") = " }{XPPEDIT 18 0 "PIECEWISE([x, x <= 0],[x^2, -x < 0 and x -2 <= 0],[8-x, 2 < x]);" "6#-%*PIECEWISEG6%7$%\"xG1F'\"\"!7$*$F'\"\"#3 2,$F'!\"\"F)1,&F'\"\"\"F,F0F)7$,&\"\")F3F'F02F,F'" }{TEXT -1 16 " , \+ then find " }{XPPEDIT 18 0 "Limit(f(x),x = 0);" "6#-%&LimitG6$-%\"fG6 #%\"xG/F)\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "Limit(f(x),x = 2 );" "6#-%&LimitG6$-%\"fG6#%\"xG/F)\"\"#" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "1 0 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }