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{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Map le Output" -1 267 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 268 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 256 "" 0 "" {TEXT 256 46 "Calculus Exploration 2: Algeb raic Limit Rules" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 267 28 "Constant Multiple Rule (CMR)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$*&%\"cG\"\"\"-%\"fG6#%\"xGF)/F-%\"aG*&F(F)- F%6$F*F.F)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 268 26 "Sum of Functions Rule (SR)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%&LimitG6$*&%!G\"\"\",&-%\"fG6#%\"xGF)-%\"gGF-F)F)/F.%\"aG,&-F%6$F+F 1F)-F%6$F/F1F)" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 270 33 "Difference of Functions Rule (DR)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$*&%!G\"\"\",&-%\"fG6#%\"xGF)-%\"gGF-!\"\"F) /F.%\"aG,&-F%6$F+F2F)-F%6$F/F2F1" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT 269 30 "Product of Functions Rule (PR)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$*&-%\"fG6#%\"xG\"\"\"-%\"g GF*F,/F+%\"aG*&-F%6$F(F/F,-F%6$F-F/F," }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 271 31 "Quotient of Functions Rule (QR)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$*&-%\"fG6#%\"xG\"\"\"-% \"gGF*!\"\"/F+%\"aG*&-F%6$F(F0F,-F%6$F-F0F/" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 257 28 "Constant Function Rule (CFR)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%-If~for~all~xG/*&%\"~G\"\" \"-%\"fG6#%\"xGF'%\"cG/*&%'~then~GF'-%&LimitG6$F(/F+%\"aGF'*&-F16$F,F3 F'%%~=~cGF'" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 258 28 "Identity Function Rule (IFR)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%-If~for~all~xG/*&%\"~G\"\"\"-%\"fG6#%\"xGF'F+/*&%'~the n~GF'-%&LimitG6$F(/F+%\"aGF'*&-F06$F+F2F'%%~=~aGF'" }}{PARA 256 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 261 21 "Power Function Rule \+ 1" }{TEXT -1 1 " " }{TEXT 266 7 "(PFR 1)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%-If~for~all~xG/*&%\"~G\"\"\"-% \"fG6#%\"xGF')F+%\"nG/**%'~then~GF'-%&LimitG6$F(/F+%\"aGF'%$~=~GF'-F26 $F,F4F')-F26$F+F4F-" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 262 29 "Power Function Rule 2 (PFR 2)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%-If~for~all~xG/*&%\"~ G\"\"\"-%\"gG6#%\"xGF')-%\"fGF*%\"nG/**%'~then~GF'-%&LimitG6$F(/F+%\"a GF'%$~=~GF'-F46$F,F6F')-F46$F-F6F/" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT 265 29 "Power Function Rule 3 (PFR 3)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%-If~for~all~xG/*&%\"~G\"\"\"-%\"hG6#% \"xGF')-%\"fGF*-%\"gGF*/**%'~then~GF'-%&LimitG6$F(/F+%\"aGF'%$~=~GF'-F 56$F,F7F')-F56$F-F7-F56$F/F7" }}{PAGEBK }{PARA 256 "" 0 "" {TEXT 277 32 "Algebraic Limit Rules: Example 1" }}{PARA 256 "" 0 "" {TEXT -1 0 " " }}{PARA 257 "" 0 "" {TEXT -1 199 "Next, we will carefully evaluate t he following limit, using the limit rules listed on the previous page, and justifying each step by labeling each one with the appropriate ru le(s) that is (are) used." }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%&Limit G6$*&%!G\"\"\",(*$)%\"xG\"\"#F)F)*&\"\"$F)F-F)F)\"\"%F)F)/F-F),(-F%6$F +F2F)-F%6$,$*&F0F)F-F)F)F2F)-F%6$F1F2F)%'~By~SRG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%!G,(*$)-%&LimitG6$%\"xG/F+\"\"\"\"\"#F-F--F)6$,$*&\" \"$F-F+F-F-F,F--F)6$\"\"%F,F-%*~By~PFR~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%!G,(\"\"\"F&-%&LimitG6$,$*&\"\"$F&%\"xGF&F&/F-F&F&-F( 6$\"\"%F.F&%(~By~IFRG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%!G,&\"\"&\" \"\"-%&LimitG6$,$*&\"\"$F'%\"xGF'F'/F.F'F'%(~By~CFRG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%!G,&\"\"&\"\"\"*&\"\"$F'-%&LimitG6$%\"xG/F-F'F'F'% (~By~CMRG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%!G\"\")%(~By~IFRG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%$So~G/-%&LimitG6$*&%!G\"\"\",(*$)%\"x G\"\"#F*F**&\"\"$F*F.F*F*\"\"%F*F*/F.F*,$*&\"\")F*F)F*F*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%jnNote~that~this~is~an~example~of~the~(by~now)~ familiar~fact~thatG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%Dif~f~is~a~co ntinuous~function~at~x~G%\"aG/*&%'~then~G\"\"\"-%&LimitG6$-%\"fG6#%\"x G/F0F%F)-F.6#F%" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 278 32 "Algebraic Limit Rules: Example 2" }{TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 200 "Again, we will caref ully evaluate the following limit, using the limit rules listed on the previous page, and justifying each step by labeling each one with the appropriate rule(s) that is (are) used." }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%&LimitG6$*&,&*$)%\"xG\"\"#\"\"\"F-F-!\"\"F-,&F+F-F-F.F./F+F,* &-F%6$F(F0F--F%6$F/F0F.%'~By~QRG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/% !G*&,&-%&LimitG6$*$)%\"xG\"\"#\"\"\"/F,F-F.-F(6$!\"\"F/F.F.,&-F(6$F,F/ F.F0F.F2%'~By~SRG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%!G,$*&\"\"$\"\" \",&-%&LimitG6$%\"xG/F-\"\"#F(-F+6$!\"\"F.F(F2F(%4~By~PFR~1,~IFR,~CFRG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%!G\"\"$%-~By~IFR,~CFRG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%$So~G/-%&LimitG6$*&,&*$)%\"xG\"\"#\"\"\"F.F .!\"\"F.,&F,F.F.F/F//F,F-,$*&\"\"$F.%!GF.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%_oNote~that~this~is~another~example~of~the~(by~now)~fa miliar~fact~thatG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%Dif~f~is~a~cont inuous~function~at~x~G%\"aG/*&%'~then~G\"\"\"-%&LimitG6$-%\"fG6#%\"xG/ F0F%F)-F.6#F%" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT 279 32 "Algebraic Limit Rules: Example 3" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 168 "In t his example, we will use the same function as in Example 2, but take t he limit as x approaches 1, instead. Notice that if we try to proceed \+ as in Example 2 we have:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%&LimitG 6$*&,&*$)%\"xG\"\"#\"\"\"F-F-!\"\"F-,&F+F-F-F.F./F+F-*&-F%6$F(F0F--F%6 $F/F0F.%*~By~QR~??G" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 402 "If we try to continue as in Example 2, we will end up with 0 in the denominator. This illustrates that we can only apply th e algebraic limit rules when all of the component limits exist, and wh en we don't introduce division by zero. So, we can't apply the Quotien t Rule in this example. This means we need a different approach to eva luate this limit (if it exists). The key is to try a little algebra: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/* (-%&LimitG6$*&,&*$)%\"xG\"\"#\"\"\"F.F.!\"\"F.,&F,F.F.F/F//F,F.F.%$~=~ GF.-F&6$*(F0F.,&F,F.F.F.F.,&%#~xGF.F.F/F/F1F.*&-F&6$F6F1F.%%~=~2GF.%1~ By~SR,~IFR,~CFRG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 181 "Notice that this is an example of a function for which t he limit exists as x approaches 1, but 1 isn't in the domain of the fu nction. What does the graph of this function look like?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 280 32 "Algebraic Limit Rules: Examp le 4" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 269 "In this example, we will use the reciprocal of the fu nction in Example 3, and take the limit as x approaches -1. In additio n, we try to use the same algebraic simplification, however in this ca se the limit does not exist. What does the graph of this function look like?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(-%&LimitG6$*&,&%\"xG\"\"\"F+!\"\"F+,&*$)F*\"\"#F+F+F+F,F,/F*F ,F+%$~=~GF+-F&6$*(F)F+,&%#~xGF+F+F,F,,&F7F+F+F+F,F1F+*&-F&6$*&F+F+,&F* F+F+F+F,F1F+%6~which~doesn't~exist.GF+" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 108 "Notice that this function diverg es in two different ways as x approaches -1 from the left or from the \+ right." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,-%&LimitG6%*&,&%\"xG\"\"\"F+!\"\"F+,&*$)F*\"\"#F+F+F+F,F,/F*F ,%%leftGF+%$~=~GF+-F&6%*(F)F+,&%#~xGF+F+F,F,,&F8F+F+F+F,F1F2F+%%~~=~GF +-F&6%*&F+F+,&F*F+F+F+F,F1F2F+,$%)infinityGF," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(-%&LimitG6%*&,&%\"xG\"\"\"F+!\"\"F+,&*$)F*\"\"#F+F+F +F,F,/F*F,%&rightGF+%$~=~GF+-F&6%*(F)F+,&%#~xGF+F+F,F,,&F8F+F+F+F,F1F2 F+*(-F&6%*&F+F+,&F*F+F+F+F,F1F2F+F3F+%)infinityGF+" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 282 33 "Algebraic Li mit Rules: Exercises " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 135 "In each exercise below, if possible, find the limit and list all the limit rules used. Otherwise, explain why the limit d oes not exist." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 1 "" {XPPMATH 20 "6$%#1.G/-%&LimitG6$\"\"#/%\"xG\"\"%%!G" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 260 "" 1 "" {XPPMATH 20 "6$%#2.G/-%&LimitG6$*$ ),&\"\"&\"\"\"*&\"\"%F,%\"xGF,!\"\"\"\"#F,/F/\"\"$%!G" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 1 "" {XPPMATH 20 "6$%#3.G/-%&LimitG6$, (*$)%\"xG\"\"#\"\"\"F-*&\"\"$F-F+F-F-\"\"(!\"\"/F+!\"%%!G" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 262 "" 1 "" {XPPMATH 20 "6$%#4.G/-%&Limit G6$,$*(\"\"$\"\"\"%\"xGF+,&F,F+\"\"%F+!\"\"F+/F,\"\"#%!G" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 263 "" 1 "" {XPPMATH 20 "6$%#5.G/-%&Limit G6$*&,&*$)%\"xG\"\"#\"\"\"F.F.F.F.,&*&\"\"$F.)F,\"\"&F.F.\"\"%F.!\"\"/ F,F5%!G" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 1 "" {XPPMATH 20 "6$%#6.G/-%&LimitG6$*&%\"xG\"\"#,&* $)F)F*\"\"\"F.F.F.!\"\"/F)\"\"!%!G" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 1 "" {XPPMATH 20 "6$%#7.G /-%&LimitG6$*&,&*$)%\"xG\"\"#\"\"\"F.F.F.F.F,!\"#/F,\"\"!%!G" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 266 " " 1 "" {XPPMATH 20 "6$%#8.G/-%&LimitG6$*&%\"xG\"\"\",&*$)F)\"\"#F*F*\" \"%!\"\"F0/F)F.%!G" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 1 "" {XPPMATH 20 "6$%#9.G/-%&LimitG6$*&%\"hG\"\"\",&F*F**&F*F*F)!\"\"F-F*/F )\"\"!%!G" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 268 "" 1 "" {XPPMATH 20 "6$%$10.G/-%&LimitG6$*&,&\"\"\"F* *&F*F*%\"hG!\"\"F-F*,&F*F**&\"\"#F*F,F-F-F-/F,\"\"!%!G" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Dr. John Pais, Mathematics Department-MICDS" }}{PARA 0 " " 0 "" {TEXT -1 45 "E-mail: pais@micds.org or pais@kinetigram.com" }} {PARA 0 "" 0 "" {TEXT -1 33 "URL: http://kinetigram.com/micds" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "A lgebraic Limit Rules" }}{EXCHG {PARA 0 "" 0 "" {TEXT 276 22 "Constant \+ Multiple Rule" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "res tart:\nc:='c':\nLimit(c*'f'(x),x=a)=c*Limit(f(x),x=a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$*&%\"cG\"\"\"-%\"fG6#%\"xGF)/F-%\"aG*& F(F)-F%6$F*F.F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 275 26 "Sum of Functions Rule (SR)" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "restart:\nLimit(``*('f'(x)+'g'(x)), x=a)=Limit('f'(x),x=a)+Limit('g'(x),x=a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$*&%!G\"\"\",&-%\"fG6#%\"xGF)-%\"gGF-F)F)/F. %\"aG,&-F%6$F+F1F)-F%6$F/F1F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 272 33 "Difference of Functions Rule (DR)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "restart:\nLimit(``*(' f'(x)-'g'(x)),x=a)=Limit('f'(x),x=a)-Limit('g'(x),x=a);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%&LimitG6$*&%!G\"\"\",&-%\"fG6#%\"xGF)-%\"gGF- !\"\"F)/F.%\"aG,&-F%6$F+F2F)-F%6$F/F2F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 273 30 "Product of F unctions Rule (PR)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "restart:\nLim it('f'(x)*'g'(x),x=a)=Limit('f'(x),x=a)*Limit('g'(x),x=a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$*&-%\"fG6#%\"xG\"\"\"-%\"gGF*F,/ F+%\"aG*&-F%6$F(F/F,-F%6$F-F/F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 274 31 "Quotient of Functions Rule (QR)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "restart:\nLimit('f'(x )/'g'(x),x=a)=Limit('f'(x),x=a)/Limit('g'(x),x=a);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%&LimitG6$*&-%\"fG6#%\"xG\"\"\"-%\"gGF*!\"\"/F+%\"a G*&-F%6$F(F0F,-F%6$F-F0F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 22 "Constant Function Rule" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "restart:\nc:='c':\nf:=x->c:\n`If f or all x`,` `*'f'(x)=f(x),\n` then `*Limit('f'(x),x=a)=Limit(f(x),x=a) *` = c`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%-If~for~all~xG/*&%\"~G\" \"\"-%\"fG6#%\"xGF'%\"cG/*&%'~then~GF'-%&LimitG6$F(/F+%\"aGF'*&-F16$F, F3F'%%~=~cGF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 259 22 "Identity Function Rule" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "restart:\nc:='c':\nf:=x->x:\n`If for all x`,` `*'f'(x)=f(x),\n` then `*Limit('f'(x),x=a)=Limit(f(x),x=a)*` = a `;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%-If~for~all~xG/*&%\"~G\"\"\"- %\"fG6#%\"xGF'F+/*&%'~then~GF'-%&LimitG6$F(/F+%\"aGF'*&-F06$F+F2F'%%~= ~aGF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 21 "Power Function Rule 1" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "restart:\nc:='c':\nn:='n':\nf:=x->x^n:\n`If for all \+ x`,` `*'f'(x)=f(x),\n#` then `*Limit('f'(x),x=a)*` = `*Limit(f(x),x=a) =Limit(x,x=a)^n*` = `*a^n;\n\n` then `*Limit('f'(x),x=a)*` = `*Limit(f (x),x=a)=Limit(x,x=a)^n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%-If~for~a ll~xG/*&%\"~G\"\"\"-%\"fG6#%\"xGF')F+%\"nG/**%'~then~GF'-%&LimitG6$F(/ F+%\"aGF'%$~=~GF'-F26$F,F4F')-F26$F+F4F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 21 "Power Functi on Rule 2" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "restar t:\nc:='c':\nn:='n':\ng:=x->f(x)^n:\n`If for all x`,` `*'g'(x)=g(x),\n ` then `*Limit('g'(x),x=a)*` = `*Limit(g(x),x=a)=Limit(f(x),x=a)^n;\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%-If~for~all~xG/*&%\"~G\"\"\"-%\"g G6#%\"xGF')-%\"fGF*%\"nG/**%'~then~GF'-%&LimitG6$F(/F+%\"aGF'%$~=~GF'- F46$F,F6F')-F46$F-F6F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 21 "Power Function Rule 3" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "restart:\nc:='c':\nn:='n ':\nh:=x->f(x)^g(x):\n`If for all x`,` `*'h'(x)=h(x),\n` then `*Limit( 'h'(x),x=a)*` = `*Limit(h(x),x=a)=Limit(f(x),x=a)^Limit(g(x),x=a);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%-If~for~all~xG/*&%\"~G\"\"\"-%\"hG6#% \"xGF')-%\"fGF*-%\"gGF*/**%'~then~GF'-%&LimitG6$F(/F+%\"aGF'%$~=~GF'-F 56$F,F7F')-F56$F-F7-F56$F/F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 "Algebraic Limit Rules: Examples" }}{EXCHG {PARA 258 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 465 "restart:\nf:=x->x^2+3*x+4:\nLimit(``*f(x ),x=1)=Limit(x^2,x=1)+Limit(3*x,x=1)+Limit(4,x=1),` By SR`;\n``=Limit( x,x=1)^2+Limit(3*x,x=1)+Limit(4,x=1),` By PFR 1`;\n``=1+Limit(3*x,x=1) +Limit(4,x=1),` By IFR`;\n``=5+Limit(3*x,x=1),` By CFR`;\n``=5+3*Limit (x,x=1),` By CMR`;\n``=8,` By IFR`;\n`So `,Limit(``*f(x),x=1)=limit(`` *f(x),x=1);\n`Note that this is an example of the (by now) familiar fa ct that`;\n`if f is a continuous function at x `=a,` then `*Limit('f'( x),x=a)='f'(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%&LimitG6$*&%!G\" \"\",(*$)%\"xG\"\"#F)F)*&\"\"$F)F-F)F)\"\"%F)F)/F-F),(-F%6$F+F2F)-F%6$ ,$*&F0F)F-F)F)F2F)-F%6$F1F2F)%'~By~SRG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%!G,(*$)-%&LimitG6$%\"xG/F+\"\"\"\"\"#F-F--F)6$,$*&\"\"$F-F+F-F -F,F--F)6$\"\"%F,F-%*~By~PFR~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%! G,(\"\"\"F&-%&LimitG6$,$*&\"\"$F&%\"xGF&F&/F-F&F&-F(6$\"\"%F.F&%(~By~I FRG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%!G,&\"\"&\"\"\"-%&LimitG6$,$* &\"\"$F'%\"xGF'F'/F.F'F'%(~By~CFRG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ /%!G,&\"\"&\"\"\"*&\"\"$F'-%&LimitG6$%\"xG/F-F'F'F'%(~By~CMRG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/%!G\"\")%(~By~IFRG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%$So~G/-%&LimitG6$*&%!G\"\"\",(*$)%\"xG\"\"#F*F**&\" \"$F*F.F*F*\"\"%F*F*/F.F*,$*&\"\")F*F)F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%jnNote~that~this~is~an~example~of~the~(by~now)~familia r~fact~thatG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%Dif~f~is~a~continuou s~function~at~x~G%\"aG/*&%'~then~G\"\"\"-%&LimitG6$-%\"fG6#%\"xG/F0F%F )-F.6#F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 442 "restart:\nf:=x->(x^2-1)/(x-1):\nLimit(f(x),x=2)=Limit(numer(f (x)),x=2)/Limit(denom(f(x)),x=2),` By QR`;\n``=(Limit(x^2,x=2)+Limit(- 1,x=2))/(Limit(x,x=2)+Limit(-1,x=2)),` By SR`;\n\n``=3/(Limit(x,x=2)+L imit(-1,x=2)),` By PFR 1, IFR, CFR`;\n``=3,` By IFR, CFR`;\n`So `,Limi t(f(x),x=2)=limit(``*f(x),x=2);\n`Note that this is another example of the (by now) familiar fact that`;\n`if f is a continuous function at \+ x `=a,` then `*Limit('f'(x),x=a)='f'(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%&LimitG6$*&,&*$)%\"xG\"\"#\"\"\"F-F-!\"\"F-,&F+F-F-F.F./F+F,* &-F%6$F(F0F--F%6$F/F0F.%'~By~QRG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/% !G*&,&-%&LimitG6$*$)%\"xG\"\"#\"\"\"/F,F-F.-F(6$!\"\"F/F.F.,&-F(6$F,F/ F.F0F.F2%'~By~SRG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%!G,$*&\"\"$\"\" \",&-%&LimitG6$%\"xG/F-\"\"#F(-F+6$!\"\"F.F(F2F(%4~By~PFR~1,~IFR,~CFRG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%!G\"\"$%-~By~IFR,~CFRG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%$So~G/-%&LimitG6$*&,&*$)%\"xG\"\"#\"\"\"F.F .!\"\"F.,&F,F.F.F/F//F,F-,$*&\"\"$F.%!GF.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%_oNote~that~this~is~another~example~of~the~(by~now)~fa miliar~fact~thatG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%Dif~f~is~a~cont inuous~function~at~x~G%\"aG/*&%'~then~G\"\"\"-%&LimitG6$-%\"fG6#%\"xG/ F0F%F)-F.6#F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 258 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 249 "restart:\nf:=x->(x^2-1)/(x-1):\na:=1:\nLimit(f(x),x= a)=Limit(numer(f(x)),x=a)/Limit(denom(f(x)),x=a),` By QR ??`;\nLimit(f (x),x=1)*` = `*Limit(factor(numer(f(x)))/denom(f(` x`)),x=1)\n=Limit(f actor(numer(f(x)))/denom(f(x)),x=1)*` = 2`,` By SR, IFR, CFR`;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/-%&LimitG6$*&,&*$)%\"xG\"\"#\"\"\"F-F -!\"\"F-,&F+F-F-F.F./F+F-*&-F%6$F(F0F--F%6$F/F0F.%*~By~QR~??G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/*(-%&LimitG6$*&,&*$)%\"xG\"\"#\"\"\"F .F.!\"\"F.,&F,F.F.F/F//F,F.F.%$~=~GF.-F&6$*(F0F.,&F,F.F.F.F.,&%#~xGF.F .F/F/F1F.*&-F&6$F6F1F.%%~=~2GF.%1~By~SR,~IFR,~CFRG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 9 "Ex ample 4" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "restart:\nf:=x->(x-1)/( x^2-1):\na:=-1:\nLimit(f(x),x=a)*` = `*Limit(numer(f(x))/factor(denom( f(` x`))),x=a)\n=Limit(numer(f(x))/factor(denom(f(x))),x=a)*` which do esn't exist.`;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(-%&LimitG6$*&,& %\"xG\"\"\"F+!\"\"F+,&*$)F*\"\"#F+F+F+F,F,/F*F,F+%$~=~GF+-F&6$*(F)F+,& %#~xGF+F+F,F,,&F7F+F+F+F,F1F+*&-F&6$*&F+F+,&F*F+F+F+F,F1F+%6~which~doe sn't~exist.GF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "Limit(f(x),x=a,left)*` = `* Limit(numer(f(x))/factor(denom(f(` x`))),x=a,left)\n*` = `*Limit(nume r(f(x))/factor(denom(f(x))),x=a,left)=limit(f(x),x=a,left);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,-%&LimitG6%*&,&%\"xG\"\"\"F+!\"\"F+,&*$)F *\"\"#F+F+F+F,F,/F*F,%%leftGF+%$~=~GF+-F&6%*(F)F+,&%#~xGF+F+F,F,,&F8F+ F+F+F,F1F2F+%%~~=~GF+-F&6%*&F+F+,&F*F+F+F+F,F1F2F+,$%)infinityGF," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 158 "Limit(f(x),x=a,right)*` = `*Limit(numer(f(x))/fact or(denom(f(` x`))),x=a,right)\n=Limit(numer(f(x))/factor(denom(f(x))), x=a,right)*` = `*limit(f(x),x=a,right);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(-%&LimitG6%*&,&%\"xG\"\"\"F+!\"\"F+,&*$)F*\"\"#F+F+F+F,F,/F*F ,%&rightGF+%$~=~GF+-F&6%*(F)F+,&%#~xGF+F+F,F,,&F8F+F+F+F,F1F2F+*(-F&6% *&F+F+,&F*F+F+F+F,F1F2F+F3F+%)infinityGF+" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "Algebraic Limit Rules:" }{TEXT 283 10 " Exercises" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 284 11 "Exercise 1." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "n:=1:\nf:=x->2:\na:=4:\n``||n||`.`,Limit(f(x) ,x=a)=``;``;\n``||n||`.`,Limit(f(x),x=a)=limit(f(x),x=a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%#1.G/-%&LimitG6$\"\"#/%\"xG\"\"%%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%#1.G /-%&LimitG6$\"\"#/%\"xG\"\"%F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 285 11 "Exercise 2." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "n:=2:\nf:=x->(5-4*x)^2: \na:=3:\n``||n||`.`,Limit(f(x),x=a)=``;``;\n``||n||`.`,Limit(f(x),x=a) =limit(f(x),x=a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%#2.G/-%&LimitG6$ *$),&\"\"&\"\"\"*&\"\"%F,%\"xGF,!\"\"\"\"#F,/F/\"\"$%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%#2.G/-%& LimitG6$*$),&\"\"&\"\"\"*&\"\"%F,%\"xGF,!\"\"\"\"#F,/F/\"\"$\"#\\" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 286 11 "Exercise 3." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "n:=3:\nf:=x->x^2+3*x-7:\na:=-4:\n``||n||`.`,Limit(f( x),x=a)=``;``;\n``||n||`.`,Limit(f(x),x=a)=limit(f(x),x=a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%#3.G/-%&LimitG6$,(*$)%\"xG\"\"#\"\"\"F-*&\" \"$F-F+F-F-\"\"(!\"\"/F+!\"%%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%#3.G/-%&LimitG6$,(*$)%\"xG\"\"#\" \"\"F-*&\"\"$F-F+F-F-\"\"(!\"\"/F+!\"%!\"$" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 287 11 "Exercise 4 ." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "n:=4:\nf:=x->3 *x/(x+4):\na:=2:\n``||n||`.`,Limit(f(x),x=a)=``;``;\n``||n||`.`,Limit( f(x),x=a)=limit(f(x),x=a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%#4.G/-% &LimitG6$,$*(\"\"$\"\"\"%\"xGF+,&F,F+\"\"%F+!\"\"F+/F,\"\"#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%#4.G/-%&LimitG6$,$*(\"\"$\"\"\"%\"xGF+,&F,F+\"\"%F+!\"\"F+/F,\"\"#F+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 288 11 "Exercise 5." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "n:=5:\nf:=x->(x^2+1)/(3*x^5+4):\na:=-1:\n``||n||`.`, Limit(f(x),x=a)=``;``;\n``||n||`.`,Limit(f(x),x=a)=limit(f(x),x=a);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%#5.G/-%&LimitG6$*&,&*$)%\"xG\"\"#\" \"\"F.F.F.F.,&*&\"\"$F.)F,\"\"&F.F.\"\"%F.!\"\"/F,F5%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%#5.G/-%& LimitG6$*&,&*$)%\"xG\"\"#\"\"\"F.F.F.F.,&*&\"\"$F.)F,\"\"&F.F.\"\"%F.! \"\"/F,F5F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 289 11 "Exercise 6." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "n:=6:\nf:=x->x^2/(x^2+1):\na:=0:\n``||n||`.` ,Limit(f(x),x=a)=``;``;\n``||n||`.`,Limit(f(x),x=a)=limit(f(x),x=a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%#6.G/-%&LimitG6$*&%\"xG\"\"#,&*$)F) F*\"\"\"F.F.F.!\"\"/F)\"\"!%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%#6.G/-%&LimitG6$*&%\"xG\"\"#,&*$) F)F*\"\"\"F.F.F.!\"\"/F)\"\"!F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 290 11 "Exercise 7." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "n:=7:\nf:=x->(x^2+1)/x^2 :\na:=0:\n``||n||`.`,Limit(f(x),x=a)=``;``;\n``||n||`.`,Limit(f(x),x=a )=limit(f(x),x=a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%#7.G/-%&LimitG6 $*&,&*$)%\"xG\"\"#\"\"\"F.F.F.F.F,!\"#/F,\"\"!%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%#7.G/-%&LimitG6 $*&,&*$)%\"xG\"\"#\"\"\"F.F.F.F.F,!\"#/F,\"\"!%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 291 11 "Exercise 8." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "n:=8:\nf:=x->x/(x^2-4):\na:=2:\n``||n||`.`,Limit(f(x),x=a)=``;``; \n``||n||`.`,Limit(f(x),x=a)=limit(f(x),x=a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%#8.G/-%&LimitG6$*&%\"xG\"\"\",&*$)F)\"\"#F*F*\"\"%!\" \"F0/F)F.%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%#8.G/-%&LimitG6$*&%\"xG\"\"\",&*$)F)\"\"#F*F*\"\"%! \"\"F0/F)F.%*undefinedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 292 11 "Exercise 9." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "n:=9:\nf:=h->h*(1-1/h):\na:=0:\n ``||n||`.`,Limit(f(h),h=a)=``;``;\n``||n||`.`,Limit(f(h),h=a)=limit(f( h),h=a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%#9.G/-%&LimitG6$*&%\"hG\" \"\",&F*F**&F*F*F)!\"\"F-F*/F)\"\"!%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%#9.G/-%&LimitG6$*&%\"hG \"\"\",&F*F**&F*F*F)!\"\"F-F*/F)\"\"!F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 293 12 "Exercise 10. " }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "n:=10:\nf:=h->( 1-1/h)/(1-2/h):\na:=0:\n``||n||`.`,Limit(f(h),h=a)=``;``;\n``||n||`.`, Limit(f(h),h=a)=limit(f(h),h=a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%$ 10.G/-%&LimitG6$*&,&\"\"\"F**&F*F*%\"hG!\"\"F-F*,&F*F**&\"\"#F*F,F-F-F -/F,\"\"!%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%$10.G/-%&LimitG6$*&,&\"\"\"F**&F*F*%\"hG!\"\"F-F*,&F *F**&\"\"#F*F,F-F-F-/F,\"\"!#F*F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2755 "restart:\nwith(plottools): \nn:=2: # Exercise number\nntxt:=convert(n,sy mbol):\n # Define function f here below \n#f:=unapply(piecewise(x<-2,2*x+9,-2u%G!#<$\"3c%pA7+;io#!#:7$$!3Ym;z43m9FF0 $\"37m;(Qll\\^#F37$$!3;LLe/$f`c#F0$\"3k)H\"pfY6HBF37$$!3PLL3K\"o]T#F0$ \"3=]+01fB\\@F37$$!3]m;Hn7\\lAF0$\"3uMY1;')Qx>F37$$!3WL$ekO9o7#F0$\"3X .J[U+YC=F37$$!3#***\\7oCA$)>F0$\"3UI@a/kfs;F37$$!3Z$=F0$\"3Y!eq T7zC_\"F37$$!3++]iDGp'o\"F0$\"3uvBLgj')z8F37$$!3Zmm;u\"HW`\"F0$\"31u?: Kt[S7F37$$!3?LL3n_J+9F0$\"3)**f*Qit'Q7\"F37$$!3q****\\sZL\\7F0$\"3++U/ )))yY***!#;7$$!3#*****\\PVt(4\"F0$\"3<-D;J/(*=))Feo7$$!3^)****\\F#R;&* !#=$\"3Ouo!4YWbv(Feo7$$!3]KLe9(3(*=)F^p$\"3#y))3ruC!\\oFeo7$$!3'[mm;k` @h'F^p$\"3\\dF;p1RWeFeo7$$!3Sjmmmvvv_F^p$\"399D584kb]Feo7$$!3(*)**\\7B 67s$F^p$\"3#zHT45V+@%Feo7$$!3?nmm;V$\"35d$\\@&\\MXGFeo7$$\"3qa+]i!o<-'F]r$\"3'\\w&*z9J\\E#Feo7$$ \"3(pLL3-$=-@F^p$\"3&\\%f\"pcL)H+]7VLA&yF^p$\"3k?WT_uFcMF07$$\"3jpm;a?@.$*F^p$\"3VX+I2D6N;F0 7$$\"3)******\\\\@-3\"F0$\"3F0cg@&e>h%F^p7$$\"3Q++v$opoA\"F0$\"3AT**>x W=g&)!#?7$$\"3c+](oMf(o8F0$\"3'GFG_>0mD#F^p7$$\"3#)***\\ii.j_\"F0$\"3a $Ri?5*\\@7F07$$\"3%GLL$oT'ym\"F0$\"3#o_I3TnPz#F07$$\"3'3++DE5!>=F0$\"3 Y=^GhGO!=&F07$$\"3Mm;a)3rf&>F0$\"3c%G2d%GKuzF07$$\"3*4++vW0d5#F0$\"3?$ ))41!4dr6Feo7$$\"3;L$3-\"QfYAF0$\"3/,H2c(=\"*e\"Feo7$$\"3C+]PWF'QR#F0$ \"3Mr0vk^Z$4#Feo7$$\"3[LL$e/Xy`#F0$\"3CPRKz<4ED7J$Feo7$$\"3%ymmm(zvLGF0$\"3Aw%[e#HE8SFeo7$$\"3-nm\"zAAA) HF0$\"38I<-b,&4![Feo7$$\"3LM$3-7d%HJF0$\"3q))Gs1Xx^cFeo7$$\"3#4++]p]ZE $F0$\"3eyZ!3e_Z\\'Feo7$$\"3$QL$e*R7)>MF0$\"3!o$p,ft$H`(Feo7$$\"3'pmmmV ,&eNF0$\"3+/t%H@'oE&)Feo7$$\"3<+](o(GP1PF0$\"3)GY&RO$GSl*Feo7$$\"3g+]7 8Z!z%QF0$\"3kMjvBu&)z5F37$$\"\"%F*$\"$@\"F*-%'COLOURG6&%$RGBG$F*F*Fa[l $\"*++++\"!\")-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%%!GF\\\\l-%%F ONTG6#%(DEFAULTG-%%VIEWG6$;F(Fiz;$!\"\"F*$\"#5F*" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*$),&\"\"&\"\"\"*&\"\"%F,F'F,!\"\"\"\"#F, " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%UCompute~the~following~limits~and~fill~in~the~blanks:G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 &/-%&LimitG6%-%\"fG6#%\"xG/F*!\"#%%leftG%&____~G/-F%6%F'F+%&rightGF./- F%6$F'F+F./-F(6#F,%%____G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&/-%&LimitG6%-%\"fG6#%\"xG/F*!\"\"%%lef tG%&____~G/-F%6%F'F+%&rightGF./-F%6$F'F+F./-F(6#F,%%____G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/-%&Limi tG6%-%\"fG6#%\"xG/F*\"\"!%%leftG%&____~G/-F%6%F'F+%&rightGF./-F%6$F'F+ F./-F(6#F,%%____G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6&/-%&LimitG6%-%\"fG6#%\"xG/F*\"\"\"%%leftG%&____~ G/-F%6%F'F+%&rightGF./-F%6$F'F+F./-F(6#F,%%____G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/-%&LimitG6%-%\" fG6#%\"xG/F*\"\"#%%leftG%&____~G/-F%6%F'F+%&rightGF./-F%6$F'F+F./-F(6# F,%%____G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/-%&LimitG6%-%\"fG6#%\"xG/F*\"\"$%%leftG%&____~G/-F%6%F 'F+%&rightGF./-F%6$F'F+F./-F(6#F,%%____G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "__________________ ___________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 33 "Filename: ExploreCalc02TexPdf.mws" }}{PARA 0 "" 0 "" {TEXT -1 36 "Copyright 2005, All Rights Reserved." }}{PARA 0 "" 0 "" {TEXT -1 44 "Permission is granted to use and modify for " }}{PARA 0 " " 0 "" {TEXT -1 37 "academic and non-commercial purposes." }}{PARA 0 " " 0 "" {TEXT -1 13 "Dr. John Pais" }}{PARA 0 "" 0 "" {TEXT -1 28 "Math ematics Department-MICDS" }}{PARA 0 "" 0 "" {TEXT -1 45 "E-mail: pais@ micds.org or pais@kinetigram.com" }}{PARA 0 "" 0 "" {TEXT -1 33 "URL: \+ http://kinetigram.com/micds" }}{PARA 0 "" 0 "" {TEXT -1 37 "_________ ____________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {MARK "128" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }