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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 18 "Limits in Calculus" }}{PARA 
258 "" 0 "" {TEXT -1 13 "(number6.mws)" }}{PARA 0 "" 0 "" {TEXT -1 
286 "The basic concept that defines \"Calculus\" it that of the \"limi
t.\"  The MAPLE command for computing the limit of some expression as \+
some variable goes to some value is     limit(expression, variable=val
ue);   Define the function f1 and perform the indicated sequence of MA
PLE commands.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {XPPEDIT 19 1 "f1:=(x^3-x^2-8*x+12)/(x^2+x-6);" "6#>%#f1G*&,**$
%\"xG\"\"$\"\"\"*$F(\"\"#!\"\"*&\"\")F*F(F*F-\"#7F*F*,(*$F(F,F*F(F*\"
\"'F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G*&,**$)%\"xG\"\"$\"\"
\"F+*$)F)\"\"#F+!\"\"*&\"\")F+F)F+F/\"#7F+F+,(F,F+F)F+\"\"'F/F/" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(x=4,f1);" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "limit(f1,x=4);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
13 "subs(x=2,f1);" }}{PARA 8 "" 1 "" {TEXT -1 43 "Error, numeric excep
tion: division by zero\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 
"limit(f1,x=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(x=-3,f1);" }}{PARA 8 "" 1 "" 
{TEXT -1 43 "Error, numeric exception: division by zero\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "limit(f1,x=-3);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#!\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 109 "To see just what is going on here, factor the numer
ator and denominator of f1 and notice the common factors. " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "factor(numer(f1));factor(denom(f1))
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"xG\"\"\"\"\"$F&F&),&F%F&\"
\"#!\"\"F*F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"xG\"\"\"\"\"$F&
F&,&F%F&\"\"#!\"\"F&" }}}{PARA 0 "" 0 "" {TEXT -1 18 "Define g as belo
w." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "g:=factor(f1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG,&%\"xG\"\"\"\"\"#!\"\"" }}}
{PARA 0 "" 0 "" {TEXT -1 96 "Does the function g equal the function f?
  To see that the answer is \"no\" perform the following:" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "Limit(g,x=2)=limit(g,x=2);subs(x=2,
g);Limit(g,x=-3)=limit(g,x=-3);subs(x=-3,g);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%&LimitG6$,&%\"xG\"\"\"\"\"#!\"\"/F(F*\"\"!" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%
&LimitG6$,&%\"xG\"\"\"\"\"#!\"\"/F(!\"$!\"&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#!\"&" }}}{PARA 0 "" 0 "" {TEXT -1 72 "Note that g is de
fined and \"continuous\" at x=2 amd x=-3 while f1 is not." }}{PARA 0 "
" 0 "" {TEXT -1 105 "Now check the limiting behavior of f1 and 1/f1 at
 infinity via the following sequence of MAPLE commands. " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Limit
(f1,x=infinity)=limit(f1,x=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/-%&LimitG6$*&,**$)%\"xG\"\"$\"\"\"F-*$)F+\"\"#F-!\"\"*&\"\")F-F+F-
F1\"#7F-F-,(F.F-F+F-\"\"'F1F1/F+%)infinityGF8" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 44 "Limit(f1,x=-infinity)=limit(f1,x=-infinity);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$*&,**$)%\"xG\"\"$\"\"\"F-*
$)F+\"\"#F-!\"\"*&\"\")F-F+F-F1\"#7F-F-,(F.F-F+F-\"\"'F1F1/F+,$%)infin
ityGF1F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Limit(1/f1,x=in
finity)=limit(1/f1,x=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%
&LimitG6$*&,**$)%\"xG\"\"$\"\"\"F-*$)F+\"\"#F-!\"\"*&\"\")F-F+F-F1\"#7
F-F1,(F.F-F+F-\"\"'F1F-/F+%)infinityG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 32 "Limit(1/f1,x=2)=limit(1/f1,x=2);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%&LimitG6$*&,**$)%\"xG\"\"$\"\"\"F-*$)F+\"\"#F-!\"
\"*&\"\")F-F+F-F1\"#7F-F1,(F.F-F+F-\"\"'F1F-/F+F0%*undefinedG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Limit(1/f1,x=2,right)=limit(
1/f1,x=2,right);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6%*&,**$
)%\"xG\"\"$\"\"\"F-*$)F+\"\"#F-!\"\"*&\"\")F-F+F-F1\"#7F-F1,(F.F-F+F-
\"\"'F1F-/F+F0%&rightG%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "Limit(1/f1,x=2,left)=limit(1/f1,x=2,left);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6%*&,**$)%\"xG\"\"$\"\"\"F-*$)F+\"
\"#F-!\"\"*&\"\")F-F+F-F1\"#7F-F1,(F.F-F+F-\"\"'F1F-/F+F0%%leftG,$%)in
finityGF1" }}}{PARA 0 "" 0 "" {TEXT -1 93 "Compute the partial fractio
n decomposition of f1 and 1/f1 and see if these limits make sense." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "convert(f1,parfrac,x);conver
t(1/f1,parfrac,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"\"\"
\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&%\"xGF$\"\"#!
\"\"F(" }}}{PARA 0 "" 0 "" {TEXT -1 30 "Now plot the graphs of f1 and \+
" }{XPPEDIT 18 0 "1/f1" "6#*&\"\"\"F$%#f1G!\"\"" }{TEXT -1 47 " to see
 if these limits make sense graphically." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "plot(f1,x=-3..3);" }}
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7N#FH7$$\"3;NLLL*zym'F1$\"3=AeTv,IU@FH7$$\"3'eLL$3N1#4(F1$\"3S`xs#RSQ'
>FH7$$\"3,pm;HYt7vF1$\"3]7R?[<)R\"=FH7$$\"37-+++xG**yF1$\"3=W@0/*>^p\"
FH7$$\"3gpmmT6KU$)F1$\"3v\\U&f\")4nd\"FH7$$\"3qNLLLbdQ()F1$\"32sy0gJ*R
[\"FH7$$\"3Z++]i`1h\"*F1$\"3o\"fbKESkR\"FH7$$\"3@-+]P?Wl&*F1$\"3D\"eE,
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%+AXESLABELSG6$Q\"x6\"Q\"yFgfl-%%VIEWG6$;F(FhelF\\gl" 1 2 0 1 10 0 2 
9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 342 "1.  For each of the fo
llowing functions compute the indicated substitutions and limits, and \+
plot the graph to see if the limiting behavior looks correct.  Recall \+
that a function is called \"continuous a x=a\" if its value at x=a equ
als its limit at x=a.  In each case say whether the function is, or is
 not, continuous at the point in question." }}{EXCHG {PARA 0 "> " 0 "
" {XPPEDIT 19 1 "f2:=(-2*x-4)/(x^3+2*x^2);" "6#>%#f2G*&,&*&\"\"#\"\"\"
%\"xGF)!\"\"\"\"%F+F),&*$F*\"\"$F)*&F(F)*$F*F(F)F)F+" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%#f2G*&,&%\"xG!\"#\"\"%!\"\"\"\"\",&*$)F'\"\"$F+F+*
&\"\"#F+)F'F1F+F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs(x = 0,f2);" }}{PARA 8 "
" 1 "" {TEXT -1 43 "Error, numeric exception: division by zero\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "limit(f2,x=0);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,$%)infinityG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 2 "In" }{TEXT 266 69 " this case, f2 is not defined at x=2, a
nd so is not continuous at x=2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 13 "subs(x=2,f2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"\"\"\"#" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 14 "limit(f2,x=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6##!\"\"\"\"#" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 106 "In this case,
 f2 is defined at 2 and the limit as x goes to 2 is the same as the va
lue of the function, so" }}{PARA 261 "" 0 "" {TEXT -1 21 "f2 is contin
uous at 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f3:=(1+h)^(1/h
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f3G),&\"\"\"F'%\"hGF'*&F'F'F(
!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 13 "subs(h=0,f3);" }}{PARA 8 "" 1 "" {TEXT 
-1 43 "Error, numeric exception: division by zero\n" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 14 "limit(f3,h=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#\"\"
\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 58 "The function is undefined \+
at 0, and so not continuous at 0" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "f4:=sin(x)/x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f
4G*&-%$sinG6#%\"xG\"\"\"F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(x=0,f4);" }
}{PARA 8 "" 1 "" {TEXT -1 43 "Error, numeric exception: division by ze
ro\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 14 "limit(f4,x=0);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 57 "The fu
nction is undefined at 0, hence not continuous at 0" }}}{PARA 0 "" 0 "
" {TEXT -1 40 "2.  Similarly consider the behavior of  " }{XPPEDIT 18 
0 "sin(1/x)" "6#-%$sinG6#*&\"\"\"F'%\"xG!\"\"" }{TEXT -1 4 "  , " }
{XPPEDIT 18 0 "x*sin(1/x)" "6#*&%\"xG\"\"\"-%$sinG6#*&F%F%F$!\"\"F%" }
{TEXT -1 8 "  , and " }{XPPEDIT 18 0 "x^2*sin(1/x)" "6#*&%\"xG\"\"#-%$
sinG6#*&\"\"\"F*F$!\"\"F*" }{TEXT -1 12 "  near x=0 ." }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 30 "f5:=piecewise(x=0,0,sin(1/x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#f5G-%*PIECEWISEG6$7$\"\"!/%\"xGF)7$-%$sinG6#*&\"\"\"F1F+!\"\"%*ot
herwiseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "limit(f5,x=0);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#;!\"\"\"\"\"" }}}{PARA 0 "" 0 "" 
{TEXT -1 34 "     a.  Plot the graph of f5 and " }{TEXT 256 28 "explai
n the MAPLE output for" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "limit(f5,x=0)
" "6#-%&limitG6$%#f5G/%\"xG\"\"!" }{TEXT -1 4 "  . " }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 31 "plot(f5, x=-1..1, y=-1.5..1.5);" }}{PARA 
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$\"#:F)" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curv
e 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 204 "Explanation: Th
e limit does not exist, but every point in the interval [-1,1] is a li
mit point of the function, i.e., for any y in [-1, 1], there are value
s of the function that are arbitarily close to y." }}{PARA 0 "" 0 "" 
{TEXT -1 9 "Continue." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "f6:
=piecewise(x=0,0,x*sin(1/x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f6
G-%*PIECEWISEG6$7$\"\"!/%\"xGF)7$*&F+\"\"\"-%$sinG6#*&F.F.F+!\"\"F.%*o
therwiseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(x=0,f6);
" }}{PARA 8 "" 1 "" {TEXT -1 43 "Error, numeric exception: division by
 zero\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Yuk! That's not what w
e wanted.  Seems to be a problem with piecewise.  Try this." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f6 := proc(x)" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 35 "if x=0 then 0; else x*sin(1/x); fi;" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
#f6Gf*6#%\"xG6\"F(F(@%/9$\"\"!F,*&F+\"\"\"-%$sinG6#*&F.F.F+!\"\"F.F(F(
F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f6(0);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "limit(f6(x),x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}
{EXCHG {PARA 263 "" 0 "" {TEXT -1 31 "The function is continuous at 0
" }{TEXT 268 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 61 "     b.  On the plot of f6 include the graphs of the line
s   " }{XPPEDIT 18 0 "y=x" "6#/%\"yG%\"xG" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "y=-x" "6#/%\"yG,$%\"xG!\"\"" }{TEXT -1 1 "." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(\{f6(x), x, -x\}, x=-1..1);" }
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91 "Since the value of the function at 0 is 0 (by inspection), the fun
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k#QrL>/R#QF37$$\"3j**\\(o*GP4wF-$\"3#e7%eL;[]JF37$$\"3=LLek6,1xF-$\"3]
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!3ozF?x6V)p%F37$$\"3F****\\P$[/a)F-$!3'o(o'*fn<:bF37$$\"31mmTN>^R')F-$
!3K84tRY\"pC'F37$Fiy$!38$=vzT!*G)oF37$$\"3Emm\"zW?)\\*)F-$!3HScp,]g$)y
F37$F^z$!3(3Jb!>_zl$)F37$$\"3Z+](oW7;@*F-$!3'fI$=]0q-%)F37$$\"3#****\\
7`f@E*F-$!3$\\>p7WI&4%)F37$$\"3N**\\i:mq7$*F-$!3Z\")>x.Ke'Q)F37$$\"3=+
+++PDj$*F-$!3#f%R-T4DM$)F37$$\"3X++voyMk%*F-$!3s!e4OwHM9)F37$Fcz$!361&
=oJM=%yF37$$\"3&****\\(=5s#y*F-$!3\\Z0#[/nt%oF37$Fhz$!35*p$*)36@SaF3-F
[[l6&F][lF^[lF^[lFa[l-%+AXESLABELSG6$Q\"x6\"Q!Fg^q-%%VIEWG6$;$F`[lF`[l
$\"\"\"F`[l%(DEFAULTG" 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" }}}}{PARA 0 "" 0 "" {TEXT 
-1 83 "Execute the following command sequences.  They are intended to \+
indicate more of the" }}{PARA 0 "" 0 "" {TEXT -1 32 "MAPLE computer al
gebra features." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f8:= (a*x
^3+c*x^2-1)/(b*x^3-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f8G*&,(*&
%\"aG\"\"\")%\"xG\"\"$F)F)*&%\"cGF))F+\"\"#F)F)F)!\"\"F),&*&%\"bGF)F*F
)F)F0F1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(x=1,f8);
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\"bGF&\"\"#F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "limit(f8
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\"F&,&%\"bGF&\"\"#F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "l
imit(f8,x=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"aG\"\"\"%
\"bG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f9:=subs(a=0,f
8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f9G*&,&!\"\"\"\"\"*&%\"cGF()
%\"xG\"\"#F(F(F(,&*&%\"bGF()F,\"\"$F(F(F-F'F'" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 21 "limit(f9,x=infinity);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f10
:=subs(b=0,f8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$f10G,(*&%\"aG\"
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "limit(f10,x=infinity);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%'signumG6#%\"aG\"\"\"%)infinityG
F)!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 49 "3.  Experiment with the funct
ion  signum(x)  and " }{TEXT 258 50 "explain what  signum(a)  means in
 this expression." }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "signum(-1); signum(0);signum(1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "plot(signum(x), x=-1..1);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7hn7$$!\"\"\"\"!F(7$$
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signum" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 238 "Explanation. \+
  signum(x) is 1 if x is postive and -1 if x is negative, and 0 (?) if
 x=0.  Thus,  -signum(a)*infinity is -infinity if a is postive, +infin
itiy if a is negative.   What happens when a=0 is not so clear from th
is expression. " }}{PARA 0 "" 0 "" {TEXT -1 4 "    " }}{PARA 0 "" 0 "
" {TEXT -1 100 "Continue. First graph the function and see if you can \+
\"guess\" the limit before computing with MAPLE." }}{EXCHG {PARA 0 "> \+
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{PARA 0 "" 0 "" {TEXT 270 28 "I would guess the limit is 0" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "limit(f11,x=infinity);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(f12, x=-2..2);" }}{PARA 13 "" 
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)F;7$$\"3oZD2Zg6<fFgt$\"3!oZD2Zg6<*F;7$$\"3,w.=m(Gp$fFgt$\"3:gP!=m(Gp$
*F;7$$\"3t@<8dK0efFgt$\"3N<sJrD`!e*F;7$$\"3XpkR!4s#yfFgt$\"3^%pkR!4s#y
*F;7$$\"3!******z)******fFgt$\"2!******z)*******Fgt-%'COLOURG6&%$RGBG$
\"*++++\"!\")$\"\"!F_bpF^bp-%'POINTSG6*7$F^bpF^bp7$$\"\"\"F_bpF^bp7$$
\"\"#F_bpF^bp7$$\"\"$F_bpF^bp7$$\"\"%F_bpF^bp7$$\"\"&F_bpF^bp7$$\"\"'F
_bpF^bpFgap-%+AXESLABELSG6$Q\"x6\"Q!Fjcp-%%VIEWG6$;F^bpFdcp%(DEFAULTG
" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C
urve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 272 22 "Guess:  Does not exis
t" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "limit(f13,x=4);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%*undefinedG" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT 273 8 "Guess: 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "
limit(f13,x=4,left);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 274 8 "Guess: 0" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 21 "limit(f13,x=4,right);" }}{PARA 11 "" 1 "" 
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deG6#,&%\"tG\"\"\"\"\"%F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
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1 0 16 "limit(f15,t=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*undefine
dG" }}}{EXCHG {PARA 266 "" 0 "" {TEXT -1 8 "Guess: 0" }}}{EXCHG {PARA 
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" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 8 "Guess:
 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "limit(f15,t=-4,right)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{PARA 257 "" 1 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Compute the following lim
its and " }{TEXT 260 129 "comment on what the results tell us about th
e relative growth rates of the two functions involved with the quotien
t in each case." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 19 "limit(x/ln(x),x=0);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 12 "
Comments.   " }{TEXT -1 13 "Unsurpising: " }{TEXT 276 1 "b" }{TEXT -1 
9 "oth x and" }{TEXT 277 1 " " }{XPPEDIT 18 0 "1/ln(x);" "6#*&\"\"\"F$
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "limit(x/ln(x),x=infinity);" 
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{XPPEDIT 18 0 "x^100;" "6#*$%\"xG\"$+\"" }{TEXT -1 1 "." }}{EXCHG 
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{TEXT -1 9 "function " }{XPPEDIT 18 0 "(x^2-y^2)/(x^2+y^2)" "6#*&,&*$%
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1 "" {XPPMATH 20 "6#/-%&LimitG6$*&,&*$)%\"xG\"\"#\"\"\"F-*$)%\"yGF,F-!
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265 51 "What does the difference quotient actually measure?" }{TEXT 
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he difference quotient as h goes to 0.  " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "g1:=x->(2*x+3)^5;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g1Gf*6#%\"xG6\"6$%)operatorG%&arrow
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{MPLTEXT 1 0 24 "dqg1:=(g1(x+h)-g1(x))/h;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%dqg1G*&,&*$),(%\"xG\"\"#*&F+\"\"\"%\"hGF-F-\"\"$F-\"
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{MPLTEXT 1 0 18 "limit(dqg1,h = 0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
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factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$),&%\"xG\"\"#\"\"$\"
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{TEXT -1 44 "Continue in the same way for g2, g3, and g4." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "g2:=x
->ln(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g2G%#lnG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "dqg2 := (g2(x+h)-g2(x))/h;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dqg2G*&,&-%#lnG6#,&%\"xG\"\"\"%\"hG
F,F,-F(6#F+!\"\"F,F-F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "l
imit(dqg2, h=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$%\"xG!\"
\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "g3:=x->1/x;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#g3Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&
\"\"\"F-9$!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "dq
g3 := (g3(x+h)-g3(x))/h;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dqg3G*&
,&*&\"\"\"F(,&%\"xGF(%\"hGF(!\"\"F(*&F(F(F*F,F,F(F+F," }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dqg3:=simplify(dqg3);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%%dqg3G,$*&\"\"\"F'*&,&%\"xGF'%\"hGF'F'F*F'!\"
\"F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "limit(dqg3, h=0);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%*$)%\"xG\"\"#F%!\"\"F*" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "g4:=x->sin(a*x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g4Gf*6#%\"xG6\"6$%)operatorG%&arrow
GF(-%$sinG6#*&%\"aG\"\"\"9$F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "dqg4 := (g4(x+h)-g4(x))/h;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%dqg4G*&,&-%$sinG6#*&%\"aG\"\"\",&%\"xGF,%\"hGF,F,F,-
F(6#*&F+F,F.F,!\"\"F,F/F3" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "limit(dqg4, h=0);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#*&-%$cosG6#*&%\"aG\"\"\"%\"xGF)F)F(F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {TEXT 
-1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 95 "Now compute the following limi
ts of the difference quotient for a general expression    G(x) . " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Limit((G(x+h)-G(x))/h,h=0)=l
imit((G(x+h)-G(x))/h,h=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&Limi
tG6$*&,&-%\"GG6#,&%\"xG\"\"\"%\"hGF.F.-F*6#F-!\"\"F.F/F2/F/\"\"!--%\"D
G6#F*F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Limit((G(x)-G(h)
)/(x-h),h=x)=limit((G(x)-G(h))/(x-h),h=x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%&LimitG6$*&,&-%\"GG6#%\"xG\"\"\"-F*6#%\"hG!\"\"F-,&F
,F-F0F1F1/F0F,--%\"DG6#F*F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 259 "" 0 "" {TEXT -1 126 "In Calculus what do we call th
e limit of the difference quotient as h goes to 0 in the first or as h
 goes to x in the second.?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "
174 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 
1 }