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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 10 "Calculus I" }}{PARA 256 "
" 0 "" {TEXT -1 28 "Lesson 21: L'Hospital's Rule" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 9 "Example 1" }
{TEXT -1 80 "\nWe find each of the following limits, first without Map
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to visualize your answer. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
21 "restart; with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the
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-$!3')*o%*ff>fm\"F-7$$\"3!)[P9y$*)))Q*F-$!3LJCSZQ45hF17$$\"3ILe!p$piZ5
!#<$\"3!3iU-7kAw%F17$$\"3U;HpDMSV6Ffq$\"3nq!RNxhHV\"F-7$$\"3%)*\\nu1D7
D\"Ffq$\"3il,g8.+2DF-7$$\"3w*\\i_V*[f8Ffq$\"3m;R!Hx03e$F-7$$\"3%**\\KX
@AQY\"Ffq$\"3Z9E'yJz-h%F-7$$\"3aT&oLPm&e:Ffq$\"3Ro%ecC$4SbF-7$$\"33Le2
UjAr;Ffq$\"35>.,;V1RmF-7$$\"3;LL?8Tmm<Ffq$\"3'\\?pa(H:kvF-7$$\"3![(oNH
4ox=Ffq$\"3\"*yU\"zy3Lj)F-7$$\"3'GLGFi\\f(>Ffq$\"3GFmN;MSt&*F-7$$\"3#
\\(o5afw$3#Ffq$\"3w?T]qH!)f5Ffq7$$\"3%\\i?zfKk=#Ffq$\"3uu)\\&)[7n:\"Ff
q7$$\"3O$3-.ZaNH#Ffq$\"3)R6+\\(z9d7Ffq7$$\"373x(*)yD>R#Ffq$\"3[.W@;^x[
8Ffq7$$\"3W;HnXD.)\\#Ffq$\"3MajWL$opW\"Ffq7$$\"3CT&=^kZ#3EFfq$\"3?,B!H
ls#[:Ffq7$$\"3eCJ=8->/FFfq$\"3_qHQK_*ej\"Ffq7$$\"3T;a7:3\"y!GFfq$\"3/Y
))ea?&*H<Ffq7$$\"3f**\\@O5'[\"HFfq$\"3]aM%=^.l#=Ffq7$$\"3Z\\Ps#z)e>IFf
q$\"3?E)y#)>k.#>Ffq7$$\"3$[(o9#G=47$Ffq$\"3xwr69%H1,#Ffq7$$\"3h\\i%RSF
MB$Ffq$\"3UX\\Q$>L-6#Ffq7$$\"3bK$GR&=_MLFfq$\"3+Hbhqj=*>#Ffq7$$\"3g*\\
iuvhCW$Ffq$\"3iUYUw_g$H#Ffq7$$\"3CT&)3`8FSNFfq$\"3kmqOAcnyBFfq7$$\"3/+
v+wI?ZOFfq$\"3mMql'*3;rCFfq7$$\"3!y?!*Q\\<yu$Ffq$\"3iGve\">(pdDFfq7$$
\"3!\\P/8b))H&QFfq$\"3U/Yre+mZEFfq7$$\"3eKeW$[7e&RFfq$\"3,>eYA$Q^t#Ffq
7$$\"3#[(=<6'fM1%Ffq$\"3i>P$)z]AEGFfq7$$\"3ummL*eOr;%Ffq$\"3cDo7VN[8HF
fq7$$\"3y;z!RFhJF%Ffq$\"34.#*o\"o^A+$Ffq7$$\"3G3-c?!3$yVFfq$\"3s!)fq&Q
G)*3$Ffq7$$\"3'***\\hgp#\\Z%Ffq$\"3qN1e!G4*pJFfq7$$\"3_$ef$pJm&e%Ffq$
\"3)>D'**\\qBhKFfq7$$\"3'omcX&pq%o%Ffq$\"3AZ+Q'f=DM$Ffq7$$\"3uuoN(G3.z
%Ffq$\"3\\pAQ)Rq(GMFfq7$$\"3#[7tsB#Q\"*[Ffq$\"3i:5Y)3M4^$Ffq7$$\"\"&\"
\"!$\"3W-!oxd7))f$Ffq-%'COLOURG6&%$RGBG$\"#5!\"\"$Fe]lFe]lF_^l-%+AXESL
ABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;$\"\"\"!\"$Fc]l%(DEFAULTG" 1 2 0 1 10 
0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 3 "h) " }{XPPEDIT 18 0 "Limit( x* e^(1/x) - x
, x = infinity)" "6#-%&LimitG6$,&*&%\"xG\"\"\")%\"eG*&F)F)F(!\"\"F)F)F
(F-/F(%)infinityG" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
8 "We see: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(x*
e^(1/x)-x,x = infinity)" "6#-%&LimitG6$,&*&%\"xG\"\"\")%\"eG*&F)F)F(!
\"\"F)F)F(F-/F(%)infinityG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Limit(  \+
x *( e^(1/x) - 1), x = infinity) " "6#-%&LimitG6$*&%\"xG\"\"\",&)%\"eG
*&F(F(F'!\"\"F(F(F-F(/F'%)infinityG" }}{PARA 0 "" 0 "" {TEXT -1 3 " = \+
" }{XPPEDIT 18 0 "Limit( (e^(1/x) - 1)/(1/x), x = infinity)  =   Limit
( (- e^(1/x)*x^(-2)) / (-x^(-2)), x = infinity)" "6#/-%&LimitG6$*&,&)%
\"eG*&\"\"\"F,%\"xG!\"\"F,F,F.F,*&F,F,F-F.F./F-%)infinityG-F%6$*&,$*&)
F**&F,F,F-F.F,)F-,$\"\"#F.F,F.F,,$)F-,$F;F.F.F./F-F1" }{TEXT -1 1 " " 
}}{PARA 0 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "Limit(e^(1/x), x = \+
infinity) = 1" "6#/-%&LimitG6$)%\"eG*&\"\"\"F*%\"xG!\"\"/F+%)infinityG
F*" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "limi
t( x * exp(1/x) - x, x = infinity);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "
plot(x * exp(1/x) - x, x = 1..100, color = red);" }{TEXT -1 0 "" }}
{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7jn7
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30]iS\"*)p[8\"F-$\"3'zgT'\\&eVg\"F-7$$\"3(\\P4r$[I-7F-$\"3y\"fSfl`(f:F
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\"F-7$$\"3****\\il&z%R:F-$\"399;&H7*=39F-7$$\"3/]7.d%\\Vn\"F-$\"39TLgW
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[r'4%F-$\"3RR'\\\")\\;E8\"F-7$$\"3y**\\P/9^N]F-$\"3pt%\\H[2i5\"F-7$$\"
3_+D1MjG\"4'F-$\"3]H_cbpw'3\"F-7$$\"3S++vj71ZrF-$\"3a5,.]!RL2\"F-7$$\"
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TiBF`q$\"3&[xpijm9-\"F-7$$\"3E++v@3%fd#F`q$\"3e&**))e,k'>5F-7$$\"3y***
\\7:Z.z#F`q$\"3-(\\Ku!\\8=5F-7$$\"3G++Dyt'p*HF`q$\"3'=$=0]1(o,\"F-7$$
\"3-+v$*Q$)f%=$F`q$\"3NVo\"R?me,\"F-7$$\"3J++D*p4xS$F`q$\"3u>I<%><[,\"
F-7$$\"3O+++(G9nf$F`q$\"3h!4PKJJS,\"F-7$$\"3s*\\PWbrl\"QF`q$\"3Q&e\"fO
f@85F-7$$\"3!)****\\*[#=6SF`q$\"30>Zm(QpD,\"F-7$$\"3P+vVH:qCUF`q$\"3^i
sU\"4H>,\"F-7$$\"3T+DJ[@-GWF`q$\"359r>2sP65F-7$$\"3i+]P%)e;SYF`q$\"3\"
o,dHI`3,\"F-7$$\"3<+v=e+)\\$[F`q$\"3YYz]pHT55F-7$$\"3'***\\7jM6X]F`q$
\"3SUYP&Qw*45F-7$$\"3-+v$R,$Qj_F`q$\"3t1:/U+c45F-7$$\"3k*\\i&*H(Q`aF`q
$\"3FG))f8\\A45F-7$$\"3E+]PLrfecF`q$\"3'eR#Q%R)))35F-7$$\"3U****\\r*)f
qeF`q$\"3Pt?([el&35F-7$$\"3W**\\()49+ygF`q$\"3#**)p<!pr#35F-7$$\"3'**
\\PM$RnyiF`q$\"3lY!3<\"f+35F-7$$\"3))**\\7dl[,lF`q$\"3%HUN18Ix+\"F-7$$
\"38****\\4Op,nF`q$\"3ms3_[!)\\25F-7$$\"3g***\\7Pda\"pF`q$\"37$Rdk:ls+
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!3-%4'o+\"F-7$$\"3x*\\PWn#=?vF`q$\"3r&*epV$ym+\"F-7$$\"3r+v=CIYGxF`q$
\"3.]3\"fe(\\15F-7$$\"3r***\\i8&4KzF`q$\"3Vmu(z2Ij+\"F-7$$\"3++v$f3z_9
)F`q$\"3$4%>PBP;15F-7$$\"3w+++n0g]$)F`q$\"3'HIuec6g+\"F-7$$\"3#)**\\iO
9dg&)F`q$\"3i\"=C4aje+\"F-7$$\"3e+vVTO!)o()F`q$\"3[?J]pPs05F-7$$\"3?++
]6u9g*)F`q$\"3%*y4'H3,c+\"F-7$$\"3Q+]7l*[%z\"*F`q$\"3'Q_YNyma+\"F-7$$
\"3#*******))[fv$*F`q$\"3r%*>:1?N05F-7$$\"3)**\\PWNFZe*F`q$\"3z+g1A[B0
5F-7$$\"3!3]i&3Q*[y*F`q$\"3%=(*f&pt705F-7$$\"$+\"F*$\"3e*y;%3n,05F--%'
COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fa_l-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%V
IEWG6$;F(Ff^l%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "i) " }
{XPPEDIT 18 0 "Limit( x ^(sin(x)), x = 0, right)" "6#-%&LimitG6%)%\"xG
-%$sinG6#F'/F'\"\"!%&rightG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Let
 " }{XPPEDIT 18 0 "y = x^(sin(x))" "6#/%\"yG)%\"xG-%$sinG6#F&" }{TEXT 
-1 8 ". Then, " }{XPPEDIT 18 0 "ln(y)  =  sin(x)* ln(x)" "6#/-%#lnG6#%
\"yG*&-%$sinG6#%\"xG\"\"\"-F%6#F,F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "
" {TEXT -1 8 "We see: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "Limit(sin(x)* ln(x), x = 0, right) = Limit( ln(x)/csc(x), x = 0, ri
ght)" "6#/-%&LimitG6%*&-%$sinG6#%\"xG\"\"\"-%#lnG6#F+F,/F+\"\"!%&right
G-F%6%*&-F.6#F+F,-%$cscG6#F+!\"\"/F+F1F2" }{TEXT -1 1 " " }}{PARA 0 "
" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "Limit( (1/x)  /  ( -csc(x) *co
t(x) ), x = 0, right)  " "6#-%&LimitG6%*(\"\"\"F'%\"xG!\"\",$*&-%$cscG
6#F(F'-%$cotG6#F(F'F)F)/F(\"\"!%&rightG" }}{PARA 0 "" 0 "" {TEXT -1 3 
" = " }{XPPEDIT 18 0 "-  Limit(sin(x)/x, x = 0, right) =  Limit(tan(x)
, x = 0, right)" "6#/,$-%&LimitG6%*&-%$sinG6#%\"xG\"\"\"F,!\"\"/F,\"\"
!%&rightGF.-F&6%-%$tanG6#F,/F,F0F1" }{TEXT -1 15 " = -1 * 0 = 0. " }}
{PARA 0 "" 0 "" {TEXT -1 7 "Hence, " }}{PARA 0 "" 0 "" {TEXT -1 1 " " 
}{XPPEDIT 18 0 "Limit(x^sin(x),x = 0,right)" "6#-%&LimitG6%)%\"xG-%$si
nG6#F'/F'\"\"!%&rightG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Limit( e^(ln
(y)), x = 0, right)  =  1" "6#/-%&LimitG6%)%\"eG-%#lnG6#%\"yG/%\"xG\"
\"!%&rightG\"\"\"" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "limit( x ^(sin(x)), x = 0, right);" }{TEXT -1 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 " plot(x ^(sin(x)), x = 0.001..2, color = red);" }
{TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&
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$\"3+6M!y40ZR)F-7$$\"39L3_<?Z[#)F1$\"3;3T(=Wn<9)F-7$$\"3mmT0V'47D\"F-$
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\"3jV*G89t*>sF-7$$\"3I;ay(ooN]#F-$\"33=/$H-oc4(F-7$$\"3#*\\(oJ7EO\"HF-
$\"3W780R*[q,(F-7$$\"3j;aV4\"3xL$F-$\"3!p<*Gv-O!)pF-7$$\"33]PRf+VgPF-$
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QA1SZ#zH(F-7$$\"31+]K?J`feF-$\"39CVV*yr4W(F-7$$\"3e;a$*RgRQiF-$\"3)Q/(
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3P'\\9hgB_-*F-7$$\"3)Q$3_liYx\"*F-$\"3BK$R1X&*4M*F-7$$\"3q$3F5ZL3d*F-$
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<$\"3.Acvt!ev.\"F^t7$$\"3%[7evj^>3\"F^t$\"3ww0O^g,s5F^t7$$\"3e;aPTuQB6
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^t7$$\"3))\\ip#*He$H\"F^t$\"3!oyj*\\B$4G\"F^t7$$\"3*HLG*Q'3SL\"F^t$\"3
Pk*=\\GCLK\"F^t7$$\"3++D'\\krrP\"F^t$\"3=^VHR-'*o8F^t7$$\"3QTNrVPG;9F^
t$\"3D/6())fB/T\"F^t7$$\"35+v])fV!f9F^t$\"3]FI?a$4cX\"F^t7$$\"3+3_^*Gx
#*\\\"F^t$\"3-d:h%*es(\\\"F^t7$$\"3'[PH%*3L8a\"F^t$\"3tW!f`pV5a\"F^t7$
$\"3;Le%4K]Ce\"F^t$\"3K3=b+5S#e\"F^t7$$\"3uuoz^i\\D;F^t$\"3b::gqbJC;F^
t7$$\"3ymmL*fapm\"F^t$\"3srK7vT-j;F^t7$$\"3`;zlZ<N4<F^t$\"3zC>$o\">f+<
F^t7$$\"3M3_=EyR^<F^t$\"3(yF5y$p]N<F^t7$$\"3/+]h0Q.!z\"F^t$\"3u*Gyn#)f
_w\"F^t7$$\"3?$e4\")*\\JM=F^t$\"3?PVRj!*H'z\"F^t7$$\"3kmmbC1#R(=F^t$\"
3(3w\"zcc-@=F^t7$$\"3!\\(=)H[[h\">F^t$\"3$REXA\\oR%=F^t7$$\"3)\\7[;$fc
c>F^t$\"3wpT.orRi=F^t7$$\"\"#\"\"!$\"3CH()>^18y=F^t-%'COLOURG6&%$RGBG$
\"*++++\"!\")$Fb[lFb[lF\\\\l-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;$\"
\"\"!\"$F`[l%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "j) " }
{XPPEDIT 18 0 "Limit( (e^x + x)^(1/x), x = infinity)" "6#-%&LimitG6$),
&)%\"eG%\"xG\"\"\"F*F+*&F+F+F*!\"\"/F*%)infinityG" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "y  =  (e^x + x)^(1/x)" "6#/%
\"yG),&)%\"eG%\"xG\"\"\"F)F**&F*F*F)!\"\"" }{TEXT -1 8 ". Then, " }
{XPPEDIT 18 0 "ln(y)  =   (1/x)* ln(e^x + x)" "6#/-%#lnG6#%\"yG*(\"\"
\"F)%\"xG!\"\"-F%6#,&)%\"eGF*F)F*F)F)" }{TEXT -1 2 ". " }}{PARA 0 "" 
0 "" {TEXT -1 8 "We see: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "Limit( ln(y), x = infinity)   =  Limit(  ln(e^x + x) / x, x = in
finity) " "6#/-%&LimitG6$-%#lnG6#%\"yG/%\"xG%)infinityG-F%6$*&-F(6#,&)
%\"eGF,\"\"\"F,F6F6F,!\"\"/F,F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 3 " = " }{XPPEDIT 18 0 "Limit(  (e^x + 1) / ( e^x + x), x = i
nfinity)  " "6#-%&LimitG6$*&,&)%\"eG%\"xG\"\"\"F+F+F+,&)F)F*F+F*F+!\"
\"/F*%)infinityG" }}{PARA 0 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "L
imit( e^x / (e^x + 1), x = infinity) = 1" "6#/-%&LimitG6$*&)%\"eG%\"xG
\"\"\",&)F)F*F+F+F+!\"\"/F*%)infinityGF+" }{TEXT -1 2 ". " }}{PARA 0 "
" 0 "" {TEXT -1 7 "Hence, " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Limit((e^x+x)^(1/x),x = infinity)" "6#-%&LimitG6$),&)%
\"eG%\"xG\"\"\"F*F+*&F+F+F*!\"\"/F*%)infinityG" }{TEXT -1 3 " = " }
{XPPEDIT 18 0 "Limit( e^(ln(y)), x = infinity) =  e" "6#/-%&LimitG6$)%
\"eG-%#lnG6#%\"yG/%\"xG%)infinityGF(" }{TEXT -1 2 ". " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "limit( (exp(x) + x)^(1/x), x = infi
nity); " }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#\"
\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot((exp(x) + x)^(
1/x), x = 2..50, color = red);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" 
{GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7\\o7$$\"\"#\"\"!$\"3
?7?5qm:kI!#<7$$\"31++]i9Rl?F-$\"3.S2tEVDUIF-7$$\"36+++DHyI@F-$\"3!G#4g
$z3=-$F-7$$\"39++](Quh>#F-$\"38T!='=7r-IF-7$$\"3A+++]echAF-$\"31Dai[E'
[)HF-7$$\"3I+++v([BR#F-$\"3IXH!yffD&HF-7$$\"3)********pJJ_#F-$\"3^Y)**
3$oECHF-7$$\"3k*****\\i9Rl#F-$\"3i]Ame=X**GF-7$$\"3u******\\vp%y#F-$\"
3gfTWOJmxGF-7$$\"3&)*****\\Z![:HF-$\"3#G&*)3f^^eGF-7$$\"3'********Rji/
$F-$\"3Zo$e$eonTGF-7$$\"35++]Z/&QF$F-$\"3***QU2Z`q\"GF-7$$\"3B+++&\\P9
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-$\"35Gv;+!o)oFF-7$$\"3E+++lC]oWF-$\"3?l'Q$QS%)[FF-7$$\"3++++SLR!)\\F-
$\"3b$4Lt!)4nt#F-7$$\"3=+++gtn&\\&F-$\"30Q(pkRP$HFF-7$$\"3N+++!Qh4,'F-
$\"3:;u7[d!\\s#F-7$$\"30+++XjzBlF-$\"3@f)G9TcAs#F-7$$\"3v******48jOqF-
$\"37%*\\Q2^m?FF-7$$\"3m******H'ev)zF-$\"3&4!\\_KS?>FF-7$$\"3e+++!z)=s
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4\\`05\"!#;$\"3yPs/upK=FF-7$$\"31+++_G'\\?\"F^s$\"3+vx*4s(H=FF-7$$\"3-
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(*******R2T/:F^s$\"36^m5CEG=FF-7$$\"3!******RC!f/;F^s$\"3\"['oj?@G=FF-
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F-7$$\"3-+++CmP&*=F^s$\"3y%p4W%=G=FF-7$$\"3(******p3u>+#F^s$\"3.AB&R$=
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3GY.cG=G=FF-7$$\"3;+++2zu&\\#F^s$\"3pO%)\\G=G=FF-7$$\"3'*******p2j(f#F
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$\"3)******\\p?d>$F^s$\"3^\"3f%G=G=FF-7$$\"3H+++e5v.LF^s$\"3_d!f%G=G=F
F-7$$\"3m*****R9@3S$F^s$\"3)*\\!f%G=G=FF-7$$\"33+++!=kW]$F^s$\"3(o/f%G
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$\"3o*****pKkwz$F^sF\\[l7$$\"31+++R([')*QF^s$\"34X!f%G=G=FF-7$$\"3!)**
****f%zt*RF^sFg[l7$$\"3y*****\\PT25%F^sFg[l7$$\"3I+++%=\"H+UF^sFg[l7$$
\"3/+++U_4-VF^sFg[l7$$\"3G+++6j0.WF^sFg[l7$$\"3A+++[!He\\%F^sFg[l7$$\"
39+++uq:-YF^sFg[l7$$\"3++++G\"esp%F^sFg[l7$$\"37+++(ob')z%F^sFg[l7$$\"
35+++*31d*[F^sFg[l7$$\"#]F*Fg[l-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fa
^l-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F(Fh]l%(DEFAULTG" 1 2 0 1 10 
0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 3 "k) " }{XPPEDIT 18 0 "Limit(  ( 1 + 1/x)^(x
^2), x = infinity)" "6#-%&LimitG6$),&\"\"\"F(*&F(F(%\"xG!\"\"F(*$F*\"
\"#/F*%)infinityG" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
4 "Let " }{XPPEDIT 18 0 "y =  ( 1 + 1/x)^(x^2)" "6#/%\"yG),&\"\"\"F'*&
F'F'%\"xG!\"\"F'*$F)\"\"#" }{TEXT -1 8 ". Then, " }{XPPEDIT 18 0 "ln(y
)  =  x^2 * ln( 1 + 1/x)" "6#/-%#lnG6#%\"yG*&%\"xG\"\"#-F%6#,&\"\"\"F.
*&F.F.F)!\"\"F.F." }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 8 "We see: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit
( x^2* ln( 1 + 1/x), x = infinity) =  Limit(  ( ln(1 + 1/x))  /  ( 1/x
^2 )  , x = infinity)" "6#/-%&LimitG6$*&%\"xG\"\"#-%#lnG6#,&\"\"\"F.*&
F.F.F(!\"\"F.F./F(%)infinityG-F%6$*&-F+6#,&F.F.*&F.F.F(F0F.F.*&F.F.*$F
(F)F0F0/F(F2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 " = " }
{XPPEDIT 18 0 "Limit( (  (-1/x^2) / ( 1 + 1/x)   )  /  (-2/x^3), x = i
nfinity)" "6#-%&LimitG6$*(,$*&\"\"\"F)*$%\"xG\"\"#!\"\"F-F),&F)F)*&F)F
)F+F-F)F-,$*&F,F)*$F+\"\"$F-F-F-/F+%)infinityG" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "Limit(  x / ( 2*(1 + \+
1/x)), x = infinity)  =  infinity" "6#/-%&LimitG6$*&%\"xG\"\"\"*&\"\"#
F),&F)F)*&F)F)F(!\"\"F)F)F./F(%)infinityGF0" }{TEXT -1 2 ". " }}{PARA 
0 "" 0 "" {TEXT -1 7 "Hence, " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Limit((1+1/x)^(x^2),x = infinity)" "6#-%&LimitG6$),&\"
\"\"F(*&F(F(%\"xG!\"\"F(*$F*\"\"#/F*%)infinityG" }{TEXT -1 3 " = " }
{XPPEDIT 18 0 "Limit(  e^(ln(y)) , x = infinity)  =  infinity" "6#/-%&
LimitG6$)%\"eG-%#lnG6#%\"yG/%\"xG%)infinityGF/" }{TEXT -1 2 ". " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "limit( ( 1 + 1/x)^(x^2), x =
 infinity);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infini
tyG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot(( 1 + 1/x)^(x^2
), x = 2..10, color = red);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" 
{GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7W7$$\"\"#\"\"!$\"3++
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y1^BpF-7$$\"3MLLLBAt'\\#F-$\"3uxEKN*RW;)F-7$$\"3aLLLjN\\oEF-$\"3T(3)pj
=)Rk*F-7$$\"3Zmmm^&Q%RGF-$\"36z;*[-L)Q6!#;7$$\"3GLLLQk#z*HF-$\"3*[&Ro`
m8H8FG7$$\"35+++l9.iJF-$\"3AgTy)3#Hg:FG7$$\"3?LLL=\"\\<L$F-$\"3gn\"4!>
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T^O^d6!e#FG7$$\"3OLLLB6@GQF-$\"3_y%z*RX()**HFG7$$\"3&)******f-w+SF-$\"
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AKLL.a#o$))F-$\"3$o*[<W*)eAVF_v7$$\"3ammm^Q40!*F-$\"3m/-v=6l6^F_v7$$\"
3y******z]rf\"*F-$\"3)o%eP>e@jfF_v7$$\"3rlmmc%GpL*F-$\"3:_9\\7z=:rF_v7
$$\"3$*)****\\LzhT*F-$\"3$=NeJ.L+q(F_v7$$\"39KLL8-V&\\*F-$\"37BC+!z$)H
L)F_v7$$\"3;mm;z\"G*z&*F-$\"3_g#=Jr#Hl!*F_v7$$\"3=+++XhUk'*F-$\"3sbnp!
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e=CLw(e6F`[l7$$\"34++]2%)38**F-$\"33t-=2&oOE\"F`[l7$$\"#5F*$\"33QA)RBh
!y8F`[l-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fg\\l-%+AXESLABELSG6$Q\"x6
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0 "" }}{PARA 0 "" 0 "" {TEXT 256 9 "Example 2" }}{PARA 0 "" 0 "" 
{TEXT -1 66 "The diagram shows a sector of a circle with central angle
 theta = " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 80 "Let A = A(theta) be the area of the regio
n between the chord PR and the arc PR. " }}{PARA 0 "" 0 "" {TEXT -1 
50 "Let B = B(theta) be the area of the triangle PQR. " }}{PARA 0 "" 
0 "" {TEXT -1 31 "Find Limit( A(theta)/B(theta), " }{XPPEDIT 18 0 "the
ta;" "6#%&thetaG" }{TEXT -1 63 " = 0, right). Plot A(theta)/B(theta) t
o visualize your answer. " }}{PARA 0 "" 0 "" {TEXT -1 52 "From your pl
ot what happens to A(theta)/B(theta) as " }{XPPEDIT 18 0 "theta;" "6#%
&thetaG" }{TEXT -1 9 " goes to " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }
{TEXT -1 5 " /2? " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Here is the \+
code for the diagram. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "d
ata:= ([0,0], [0.5, sqrt(3)/2]):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "data1:= ([0.5, sqrt(3)/2], [1,0]):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 35 "data2:= ([0.5,sqrt(3)/2], [0.5,0]):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "a:= plot( sqrt(1 - x^2), x =
 0..1, color = red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "b:=
 plot([data], color = orange):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 34 "c:= plot([data1], color = orange):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 34 "d:= plot([data2], color = orange):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "e:= textplot([.07,.05,`O`],color = \+
magenta):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "f:= textplot([
.54,.05,`Q`], color = magenta):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 44 "g:= textplot([.9,.05,`R`], color = magenta):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 43 "h:= textplot([.4,.8,`P`], color = magenta
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "l:= textplot([.25,.15
,`theta`], color = magenta):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 41 "m:= textplot([.65,.35,`B`], color = red):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 39 "n:= textplot([.8,.5,`A`], color = red):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "o:= plot(sqrt(.04 - x^2), x \+
= 0.1..0.2, color = red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
54 "display(a,b,c,d,e,f,g,h,l,m,n,o, scaling=constrained);" }}{PARA 
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\">9)p0<'R'F/7$$\"35+++dg6<>F2$\"3;Q#R#4s%zp&F/7$$\"32+++w(Gp$>F2$\"3S
)eUUSSI)\\F/7$$\"37+++oK0e>F2$\"3c-&**G*fkuSF/7$$\"3-+++&oi\"o>F2$\"3i
pN0%fsVb$F/7$$\"3#******>5s#y>F2$\"3L6)p2Of+%HF/7$$\"3#******f2/P)>F2$
\"3oFv)>#Q!za#F/7$$\"3++++^g8*)>F2$\"3)***pswDx\"3#F/7$$\"35+++E!oX*>F
2$\"3mS#)pEt.t9F/7$$\"\"#Fg_lF(Fb\\l-%+AXESLABELSG6%Q\"xFi^lQ!6\"%(DEF
AULTG-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;F(F*Ffbm" 1 2 0 1 10 0 2 9 
1 4 1 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" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Let the \+
radius of the circle be r. We have: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {XPPEDIT 18 0 "A(theta)" "6#-%\"AG6#%&thetaG" }
{TEXT -1 50 " = area of entire sector - area of triangle OPR = " }
{XPPEDIT 18 0 "theta*r^2/2  - (r /2)*(r* sin(theta)) = (r^2/2)*( theta
- sin(theta) )" "6#/,&*(%&thetaG\"\"\"*$%\"rG\"\"#F'F*!\"\"F'*(F)F'F*F
+*&F)F'-%$sinG6#F&F'F'F+*(F)F*F*F+,&F&F'-F/6#F&F+F'" }{TEXT -1 1 " " }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "B(theta
)" "6#-%\"BG6#%&thetaG" }{TEXT -1 17 " = |QR| |PQ|/2 = " }{XPPEDIT 18 
0 "(r /2) *(1 - cos(theta)) (rsin(theta))" "6#*(%\"rG\"\"\"\"\"#!\"\"-
,&F%F%-%$cosG6#%&thetaGF'6#-%%rsinG6#F-F%" }{TEXT -1 3 " . " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Hence, " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "Limit( \+
A(theta)/B(theta),  theta= 0, right) = Limit( ( (r^2 /2)*( theta- sin(
theta) ) ) / ( (r^2 /2)*(1 - cos(theta) )*sin(theta) ),  theta= 0, rig
ht) " "6#/-%&LimitG6%*&-%\"AG6#%&thetaG\"\"\"-%\"BG6#F+!\"\"/F+\"\"!%&
rightG-F%6%**%\"rG\"\"#F8F0,&F+F,-%$sinG6#F+F0F,**F7F8F8F0,&F,F,-%$cos
G6#F+F0F,-F;6#F+F,F0/F+F2F3" }}{PARA 0 "" 0 "" {TEXT -1 3 " = " }
{XPPEDIT 18 0 "Limit( ( 1 - cos(theta) ) / ( (1 - cos(theta)) *cos(the
ta) + sin(theta) *sin(theta) ), theta= 0, right)" "6#-%&LimitG6%*&,&\"
\"\"F(-%$cosG6#%&thetaG!\"\"F(,&*&,&F(F(-F*6#F,F-F(-F*6#F,F(F(*&-%$sin
G6#F,F(-F76#F,F(F(F-/F,\"\"!%&rightG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "
" {TEXT -1 3 " = " }{XPPEDIT 18 0 "Limit( ( 1 - cos(theta) ) / ( cos(t
heta) - cos(theta)^2 + sin(theta)^2 ), theta= 0, right)" "6#-%&LimitG6
%*&,&\"\"\"F(-%$cosG6#%&thetaG!\"\"F(,(-F*6#F,F(*$-F*6#F,\"\"#F-*$-%$s
inG6#F,F4F(F-/F,\"\"!%&rightG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 3 " = " }{XPPEDIT 18 0 "Limit( sin(theta) / ( -sin(theta) + 4
 *sin(theta)* cos(theta) ), theta= 0, right) " "6#-%&LimitG6%*&-%$sinG
6#%&thetaG\"\"\",&-F(6#F*!\"\"*(\"\"%F+-F(6#F*F+-%$cosG6#F*F+F+F//F*\"
\"!%&rightG" }}{PARA 0 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "Limit(
1/(-1+4*cos(theta)),theta = 0,right) = 1/3;" "6#/-%&LimitG6%*&\"\"\"F(
,&F(!\"\"*&\"\"%F(-%$cosG6#%&thetaGF(F(F*/F0\"\"!%&rightG*&F(F(\"\"$F*
" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "A:= x -> (
0.5) * x^2 * ( x - sin(x));" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AGR6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&)9$\"\"#\"
\"\",&F/F1-%$sinG6#F/!\"\"F1$\"\"&F6F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 48 "B:= x -> (0.5) * x * ( 1 - cos(x)) * x * sin(x);" }
{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BGR6#%\"xG6\"6$%)o
peratorG%&arrowGF(,$*()9$\"\"#\"\"\",&F1F1-%$cosG6#F/!\"\"F1-%$sinGF5F
1$\"\"&F6F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "az:=plot
(A(x)/B(x), x = 0.01..Pi/2, color = red):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 52 "bz:= plot([x,1/3, x = 0.01..Pi/2], color = magenta)
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display((az,bz));" }
{TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&
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h[l7$$\"3%e#on5SB>5FauFh[l7$$\"3#Ql&p#*yz]5FauFh[l7$$\"3Qsk#R2*\\%3\"F
auFh[l7$$\"3p.?5`t.:6FauFh[l7$$\"3I8uXePU[6FauFh[l7$$\"3?VM.kx$)z6FauF
h[l7$$\"3,jYR^Wn77FauFh[l7$$\"3=f-H:$yZC\"FauFh[l7$$\"3o$pdb3)Qy7FauFh
[l7$$\"3u$=T*)Ge2J\"FauFh[l7$$\"3=qGB(\\hQM\"FauFh[l7$$\"3C_kC%f!pw8Fa
uFh[l7$$\"3@9Q!\\7doS\"FauFh[l7$$\"3A*QGLPJ9W\"FauFh[l7$$\"3'o$>G$*\\N
s9FauFh[l7$$\"3LY)f>-E`]\"FauFh[l7$$\"3!*pIT.N)o`\"FauFh[l7$$\"3/+++Fj
zq:FauFh[l-Fjz6&F\\[lF][lF`[lF][l-%+AXESLABELSG6%Q\"x6\"Q!6\"%(DEFAULT
G-%%VIEWG6$;$\"\"\"!\"#$\"+Fjzq:!\"*Fcel" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 27 "evalf( A(Pi/2) / B(Pi/2) );" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$\"*Fjzq&!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 29 "From the plot we see that as " }{XPPEDIT 18 0 "theta;" "6#%&the
taG" }{TEXT -1 9 " goes to " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 
6 "/2, A(" }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 4 ")/B(" }
{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 24 ") goes to approximat
ely " }{XPPEDIT 18 0 ".570796327;" "6#$\"*Fjzq&!\"*" }{TEXT -1 2 ". " 
}}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 
1 2 33 1 1 }