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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 268 17 "Lab # 1:   Limits" }}
{PARA 259 "" 0 "" {TEXT -1 75 "In this lab you we will investigate lim
its.  These come in three flavors:  " }{TEXT 269 16 "Infinite Limits,
" }{TEXT -1 88 " when the function goes to positive or negative infini
ty as x goes to a finite number,  " }{TEXT 270 19 "Limits at Infinity,
" }{TEXT -1 135 " this is often referred to as \"end behavior\", and d
escribes the limit of the function as x goes to positive or negative i
nfinity, and  " }{TEXT 271 14 "Finite Limits," }{TEXT -1 72 " when a f
unction goes to a finite number as x goes to a finite number.  " }}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 38 "Infinite Limits and Limits at Inf
inity" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Here we  plot the functio
n f(x) =  " }{XPPEDIT 18 0 "3*x/(x-2);" "6#*(\"\"$\"\"\"%\"xGF%,&F&F%
\"\"#!\"\"F)" }{TEXT -1 127 "  which has a discontinuity at x = 2. Spe
cifically, the function is undefined at this value.  We first define t
he function by  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := x \+
-> 3*x/(x-2);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG
f*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*(\"\"$\"\"\"9$F/,&F0F/\"\"#!\"\"
F3F/F(F(F(" }}}{EXCHG {PARA 0 "" 1 "" {TEXT -1 136 "Now let's try to p
lot this with the plot command.  Try the following three options seque
ntially to get a better picture of this graph.  " }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 36 "plot(f(x),x=-4..8,labels=[\"x\",\"f\"]);" }
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "This shows some peculiar behavior
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s a discontinuity with the following command.  You need not specify wh
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0 "" 0 "" {TEXT -1 108 "The above sequence of graphs should make it cl
ear that the limit of f(x) as x approaches 2 from the left is " }}
{PARA 0 "" 0 "" {TEXT -1 20 "negative infinity (-" }{XPPEDIT 18 0 "inf
inity;" "6#%)infinityG" }{TEXT -1 82 ")  and the limit of f(x) as x ap
proaches 2 from the right is positive infinity  ( " }{XPPEDIT 18 0 "in
finity;" "6#%)infinityG" }{TEXT -1 121 ").  These are called infinite \+
limits and describe an infinite limit of the function as x approaches \+
a finite value (2).  " }}{PARA 0 "" 0 "" {TEXT -1 6 " Does " }
{XPPEDIT 18 0 "limit(f(x),x = 2);" "6#-%&limitG6$-%\"fG6#%\"xG/F)\"\"#
" }{TEXT -1 48 "  exist?  Try f (1.99999)  and  f (2.00001).    " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Maple has a command for finding th
ese limits. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "limit(f(x),
x=2);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*undefinedG" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 250 "This \"undefined\" makes sense
 because the limit from the left is different from the limit from the \+
right.  We can determine these limits from the left and right by inclu
ding this in the limit command. The command below finds the limit from
 the left.  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "limit(f(x),
x=2,left);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%)infinityG!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Similarly the limit from the right
 of x = 2 is found by " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "l
imit(f(x),x=2,right);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 249 "Therefore we say: \"the limit \+
of f as x approaches 2 from the left is negative infinity and the limi
t of f as x approaches 2 from the right is (positive) infinity. Becaus
e these two are not the same, \"the limit of f as x approaches 2 is un
defined\".  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 222 "It also appears that as x goes to negative and positive infini
ty, the function tends towards a finite value.  I initially guess this
 value to be 3.   I check this by increasing my domain and plotting th
e line y=3 as well. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "plo
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ve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 204 "It is now cle
ar that as x goes to positive or negative infinity, the function tends
 to 3.  These are called limits at infinity. You state: \"The limit as
 x approaches positive or negative infinity is 2\".  " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 79 "Problem number 1 asks for these same type
s of limits for a different function. " }}}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 13 "Finite Limits" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "For \+
most functions " }{XPPEDIT 18 0 "limit(f(x),x = c);" "6#-%&limitG6$-%
\"fG6#%\"xG/F)%\"cG" }{TEXT -1 25 "  =  f(c), provided that " }
{XPPEDIT 18 0 "c;" "6#%\"cG" }{TEXT -1 21 " is in the domain of " }
{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 93 " .  With that said I will i
nstead demonstrate several limits that exist but are not given by " }
{XPPEDIT 18 0 "f(c);" "6#-%\"fG6#%\"cG" }{TEXT -1 49 ".  We will demon
strate these limits graphically. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
272 3 "(1)" }{TEXT -1 9 " I Claim " }{XPPEDIT 18 0 "limit(sin(x)/x,x =
 0);" "6#-%&limitG6$*&-%$sinG6#%\"xG\"\"\"F*!\"\"/%\"xG\"\"!" }{TEXT 
-1 106 "  =  1.  Notice you can not put zero in for x. You get the ind
eterminate form (0/0). Let's first plot it. " }}}{EXCHG {PARA 0 "> " 
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os3N%***F*7$$\"3U(***\\i!*3`iF0$\"3?r*e,U%[$***F*7$$\"3SMLLL*zym'F0$\"
35!*QJV:f#***F*7$$\"3?KLL3N1#4(F0$\"3#**f+Q@>;***F*7$$\"3Ynm;HYt7vF0$
\"3=(Q92z&f!***F*7$$\"3!*)******p(G**yF0$\"3W%pfJX.'*)**F*7$$\"30nmmT6
KU$)F0$\"3Ws3!=)\\S))**F*7$$\"3[JLLLbdQ()F0$\"375KWTxF()**F*7$$\"3/,+]
i`1h\"*F0$\"3c&)ee[$=g)**F*7$$\"3M****\\P?Wl&*F0$\"3AJ'R9OdZ)**F*7$$\"
3/+++++++5F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fa[lF`[l-%+AXESLABELSG
6$Q\"x6\"Q!Ff[l-%%VIEWG6$;$F_[lF_[l$\"\"\"F_[l%(DEFAULTG" 1 2 0 1 10 
0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 37 "Its pretty clear from the graph that " }
{XPPEDIT 18 0 "limit(sin(x)/x,x = 0);" "6#-%&limitG6$*&-%$sinG6#%\"xG
\"\"\"F*!\"\"/F*\"\"!" }{TEXT -1 47 "  =  1.  Let's have Maple determi
ne the limit. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "limit(sin
(x)/x, x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 47 "Again further evidence that the claim is \+
true. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 273 5 "(2)  " }{TEXT -1 11 "Th
e number " }{XPPEDIT 18 0 "e;" "6#%\"eG" }{TEXT -1 15 " is defined by \+
" }{XPPEDIT 18 0 "limit((1+x)^(1/x),x = 0);" "6#-%&limitG6$),&\"\"\"F(
%\"xGF(*&F(F(F)!\"\"/%\"xG\"\"!" }{TEXT -1 5 "  =  " }{XPPEDIT 18 0 "e
;" "6#%\"eG" }{TEXT -1 79 ".  Let's verify this limit exists and appro
ximate its value. First we plot y = " }{XPPEDIT 18 0 "(1+x)^(1/x)" "6#
),&\"\"\"F%%\"xGF%*&F%F%F&!\"\"" }{TEXT -1 29 " over a domain containi
ng 0. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot((1+x)^(1/x),
x=-0.5..0.5);" }}{PARA 13 "" 1 "" {GLPLOT2D 353 353 353 {PLOTDATA 2 "6
%-%'CURVESG6$7S7$$!3++++++++]!#=$\"\"%\"\"!7$$!3MLLLe%G?y%F*$\"3?diTD^
<(*Q!#<7$$!3OmmT&esBf%F*$\"3n*Q()G$*HS\"QF37$$!3KLL$3s%3zVF*$\"33it$[$
4!os$F37$$!33LL$e/$QkTF*$\"3\"RVV0A<]k$F37$$!3!pm;/\"=q]RF*$\"3Q0$=K72
!pNF37$$!3SLL3_>f_PF*$\"35\\$*>m,(G]$F37$$!3))***\\(o1YZNF*$\"3,<$G%e!
y$QMF37$$!3]LL3-OJNLF*$\"3y*yd/zobP$F37$$!3C++v$*o%Q7$F*$\"3EsR;@*[lJ$
F37$$!3ammm\"RFj!HF*$\"3#3YlVl^#fKF37$$!3JLL$e4OZr#F*$\"3(3ibK1:9@$F37
$$!3=+++v'\\!*\\#F*$\"3Q3`VrWFgJF37$$!33+++DwZ#G#F*$\"3-m8>(Q*e6JF37$$
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=t)eC\"F*$\"3z%pA]Ir&4HF37$$!39nmm;1J\\5F*$\"3;>aHfd9wGF37$$!3&y***\\(
=[jL)!#>$\"3#HF)4^C*4%GF37$$!3M****\\iXg#G'F[r$\"3f\\Di&)H!*3GF37$$!3W
lmmT&Q(RTF[r$\"3Gq(e;tpnx#F37$$!3;nm;/'=><#F[r$\"3?zUn>9S[FF37$$!3vDML
Le*e$\\!#@$\"3!**=L**)H&*=FF37$$\"3[em;zRQb@F[r$\"3+S\\TaUb*o#F37$$\"3
'[***\\(=>Y2%F[r$\"37L.6&4$*[m#F37$$\"3Qhmm\"zXu9'F[r$\"3i7/)='p<REF37
$$\"3]'******\\y))G)F[r$\"3a:I?<tb8EF37$$\"3'*)***\\i_QQ5F*$\"3=XgY+YP
*e#F37$$\"3@***\\7y%3T7F*$\"371C?r>wmDF37$$\"35****\\P![hY\"F*$\"3cb%o
)3:^UDF37$$\"3kKLL$Qx$o;F*$\"38#3m1,^9_#F37$$\"3!)*****\\P+V)=F*$\"3o'
3Y^Z(o*\\#F37$$\"3?mm\"zpe*z?F*$\"3a3\"[=:!e![#F37$$\"3%)*****\\#\\'QH
#F*$\"3)=^Qy2E.Y#F37$$\"3GKLe9S8&\\#F*$\"3$)Rk&HJY=W#F37$$\"3R***\\i?=
bq#F*$\"3K%GkUO+JU#F37$$\"3\"HLL$3s?6HF*$\"3_.Jh26J0CF37$$\"3a***\\7`W
l7$F*$\"3o8l*fhJsQ#F37$$\"3#pmmm'*RRL$F*$\"3/!*341;KqBF37$$\"3Qmm;a<.Y
NF*$\"33da2)3;NN#F37$$\"3=LLe9tOcPF*$\"3O7'=2K:tL#F37$$\"3u******\\Qk
\\RF*$\"3UxNeT,#GK#F37$$\"3CLL$3dg6<%F*$\"3'oPri,\\mI#F37$$\"3ImmmmxGp
VF*$\"3'>K*)47pDH#F37$$\"3A++D\"oK0e%F*$\"3op?Vw(RzF#F37$$\"3A++v=5s#y
%F*$\"3au;R'4$HkAF37$$\"3++++++++]F*$\"3+++++++]AF3-%'COLOURG6&%$RGBG$
\"#5!\"\"$F-F-Fb[l-%+AXESLABELSG6$Q\"x6\"Q!Fg[l-%%VIEWG6$;$!\"&Fa[l$\"
\"&Fa[l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Certai
nly looks like the limit exists. Have Maple evaluate it" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "limit((1+x)^(1/x),x=0);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%$expG6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 179 "How about that? Maple already knows about this limit.  A
s a check, find the difference between e = exp(1) in Maple and (1+x)^(
1/x) evaluated at x = .0001.  They should be close.  " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "a := evalf((1+.0001)^(1/.0001));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG$\"+Ff9=F!\"*" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 19 "b := evalf(exp(1));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"bG$\"+G=G=F!\"*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "abs(a-b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"',f8!
\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "# end of section" }}
}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "Assignment " }}{PARA 0 "" 0 "" 
{TEXT -1 145 "Put your name, course and section in the upper left corn
er.  You will hand in two graphs (one page) with a little information \+
about each graph.  " }}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 
257 11 "Graph # 1: " }}{PARA 258 "" 0 "" {TEXT -1 23 "Consider  the fu
nction " }{XPPEDIT 18 0 "(1+2*abs(x))/x;" "6#*&,&\"\"\"F%*&\"\"#F%-%$a
bsG6#%\"xGF%F%F%F+!\"\"" }{TEXT -1 63 "  .  Note: the absolute value f
unction is called  by \"abs(x)\". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }
}{PARA 0 "" 0 "" {TEXT -1 188 "(a) Display the 2 limits at infinity, a
nd the 2 infinite limits of the function by plotting it over an approp
riate interval.  Display only one graph which captures all of these fe
atures.  " }{TEXT 265 5 "Hint:" }{TEXT -1 66 "  See the graphing comma
nds for the final graph in the section on " }{TEXT 266 38 "Infinite Li
mits and Limits at Infinity" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "(b) Describe each of the
 limits using mathematical terms.  You can get the editor to type limi
t notation by clicking on the " }{XPPEDIT 18 0 "Sigma;" "6#%&SigmaG" }
{TEXT -1 209 " button.  A small box appears in the context bar.  You c
an then type in the command using Maple notation such as \"limit(f(x),
x=0,left)\", \"enter\",  and the mathematical expression described by \+
this command:  \" " }{XPPEDIT 18 0 "limit(f(x),x = 0,left);" "6#-%&lim
itG6%-%\"fG6#%\"xG/F)\"\"!%%leftG" }{TEXT -1 256 "\"   is printed at t
he cursor position. If you can't get this to work, you can always expr
ess the limit as a statement like: \"the limit of f(x) as x goes to (n
egative) infinity is ...\" and \"the limit of f(x) as x go to zero fro
m the right (left) is ...\" .  " }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 0 "" }{TEXT 259 0 "" }
{TEXT 260 0 "" }{TEXT 261 9 "Graph #2:" }}{PARA 0 "" 0 "" {TEXT -1 11 
"Let g(x) = " }{XPPEDIT 18 0 "abs(x);" "6#-%$absG6#%\"xG" }{TEXT -1 
10 ",  h(x) = " }{XPPEDIT 18 0 "-abs(x);" "6#,$-%$absG6#%\"xG!\"\"" }
{TEXT -1 14 ", and f(x) =  " }{XPPEDIT 18 0 "x*sin(1/x);" "6#*&%\"xG\"
\"\"-%$sinG6#*&F%F%F$!\"\"F%" }{TEXT -1 58 ".  Plot these functions on
 the same axes over the domain [" }{XPPEDIT 18 0 "-Pi,Pi;" "6$,$%#PiG!
\"\"F$" }{TEXT -1 2 "]." }}{PARA 0 "" 0 "" {TEXT -1 60 "Make a convinc
ing argument, other that just the graph, that " }{XPPEDIT 18 0 "limit(
x*sin(1/x),x = 0);" "6#-%&limitG6$*&%\"xG\"\"\"-%$sinG6#*&F(F(F'!\"\"F
(/%\"xG\"\"!" }{TEXT -1 8 "  =  0. " }}}}}{MARK "3" 0 }{VIEWOPTS 1 1 
0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }