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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 40 "Limits in Maple\nMarch 2
006\nOctober 2007\n" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 270 "In this worksheet  we shall compute limits in maple.  We
 begin by examining three basic strategies to calculate limits.  Then \+
we shall investigate one-sided limits and limits involving infinity, a
nd finally we shall use limits to compute the slope of a curve at a po
int." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Basic Limits" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 72 "Let us look at several strategies to calculate a limit fo
r the example  " }{XPPEDIT 18 0 "limit(ln(x^2)/(x-1),x = 1);" "6#-%&li
mitG6$*&-%#lnG6#*$%\"xG\"\"#\"\"\",&F+F-F-!\"\"F//F+F-" }{TEXT -1 7 " \+
  .  \n" }{TEXT 262 17 "1. Direct Method." }}{PARA 0 "" 0 "" {TEXT -1 
41 "Maple's limit command has the structure  " }{TEXT 261 25 "limit( e
xpression, point)" }{TEXT -1 32 ".  In our example we would enter" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "limit(ln(x^2)/(x-1), x= 1);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "Maple found a result, but it does
 not display the problem itself, and possible input errors are hard to
 detect.\n\n" }{TEXT 263 34 "2. Two-Step Method (Recommended). " }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 106 "Here we first enter the
 inert form of the limit command (upper case 'L'), and then we ask for
 its value.  " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Limit(ln(x^2)/(x-1
), x=1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "The result is identical, but the \+
expression was displayed as well, which makes it much easier to catch \+
potential typing errors.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT 264 32 "3. Define Expression Explicitly." }}{PARA 0 "" 
0 "" {TEXT -1 99 "Finally, we can also define the expression explicitl
y and then take the limit of this expression.  " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "Y := ln(x^2)/(x-1);\nlimit(Y,x=1);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 19 "or in function form" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "f := x -> ln(x^2)/(x-1);\nlimit(f(x),x=1);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 265 13 "More Examples" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{MPLTEXT 1 0 34 "Limit( sin(x)/x, x = 0);\nvalue(%);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "Limit( (3^x-2^x)/x, x=0);
\nvalue(%);  # Compute the Limit\nevalf(%);  # Evaluation in decimal n
otation" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "We can't expect maple \+
to find a limit, which doesn't exist:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 32 "Limit( sin(1/x),x=0);\nvalue(%); " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "Limit( abs(x)/x , x=0);\nvalue(%);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 "One-Sided Limits" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 32 "For one-sided limits we need to " }{TEXT 267 17 "a
dd the direction" }{TEXT -1 38 " as an option in the limit statement. \+
" }}{PARA 0 "" 0 "" {TEXT 268 9 "Examples:" }}{PARA 0 "" 0 "" {TEXT 
-1 108 "Here we look at a function which is defined by two different f
ormulas.  A graph will clarify the situation. " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "f := x -> piecewise( x <2, x^2-2, x+1);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "plot(f,0..4,discont=true);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 36 "Now let's take the one-sided limits." }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 31 "Limit(f(x),x=2,left); value(%);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 32 "Limit(f(x),x=2,right); value(%);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "limit(f(x),x=2);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 36 "The next example is straightforward." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Limit(abs(x)/x,x=0,right);value(%);
  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Limit(abs(x)/x,x=0,le
ft);value(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "The limit as suc
h does not exist, since the one-sided limits do not match." }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 29 "Limit(abs(x)/x,x=0);value(%);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
-1 25 "Limits Involving Infinity" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Taking limits \+
as the variable approaches infinity is easily accomplished, just enter
  " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 18 "  as 'inf
inity'.  " }}{PARA 0 "" 0 "" {TEXT 269 9 "Examples:" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 39 "Limit( arctan(x),x=infinity);\nvalue(%);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Limit( x + sqrt(x^2 + 6*x +3
) , x=-infinity); value(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "Ma
ple uses the infinity symbol if the expressions are unbounded and the \+
limit is infinite.  \n" }{TEXT 270 9 "Examples:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "Limit(exp(x),x=infinity);\nvalue(%);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Limit(ln(x),x=0);\nvalue(%);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Limits of the tangent function at \+
 " }{XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 3 "  :
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Limit( tan(x), x= Pi/2);\nvalue
(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Limit( tan(x), x= P
i/2, left);\nvalue(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "L
imit( tan(x), x= Pi/2, right);\nvalue(%);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 39 "Compute t
he Slope at a Point as a Limit" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "
We consider an example.  Let  " }{XPPEDIT 18 0 "f(x) = x^2*sin(x);" "6
#/-%\"fG6#%\"xG*&F'\"\"#-%$sinG6#F'\"\"\"" }{TEXT -1 61 "  and answer \+
the following questions \n(a) Find the slope at  " }{XPPEDIT 18 0 "x =
 Pi/2;" "6#/%\"xG*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 66 "  .  \n(b) Fin
d a formula for the slope at any point (x,f(x)).     " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 271 9 "Solution:" }{TEXT -1 
30 "  First we define the function" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
30 "f := x -> x^2*sin(x); f(Pi/2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 42 "Here is a graph to illustrate the problem:" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 58 "plot([f(x), Pi*(x-Pi/2)+f(Pi/2)],x=0..3,color=[blue
,red]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 9 "Part (a):" }{TEXT -1 
29 " The difference quotient     " }{XPPEDIT 18 0 "(f(x)-f(Pi/2))/(x-P
i/2);" "6#*&,&-%\"fG6#%\"xG\"\"\"-F&6#*&%#PiGF)\"\"#!\"\"F/F),&F(F)*&F
-F)F.F/F/F/" }{TEXT -1 62 "     is the slope of the line segment conne
cting the points ( " }{XPPEDIT 18 0 "Pi/2,f(Pi/2);" "6$*&%#PiG\"\"\"\"
\"#!\"\"-%\"fG6#*&F$F%F&F'" }{TEXT -1 94 ")  and (x,f(x)).  We first d
efine the difference quotient and call it DQ, then we simplify it." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "DQ := (f(x)-f(Pi/2))/(x-Pi/2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 47 "The limit will result in the slope at the
 point" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Limit(DQ, x=Pi/2);\nvalue
(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Hence, the slope at the p
oint   ( " }{XPPEDIT 18 0 "Pi/2,Pi^2/4;" "6$*&%#PiG\"\"\"\"\"#!\"\"*&F
$F&\"\"%F'" }{TEXT -1 7 " )  is " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }
{TEXT -1 3 ".  " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
260 9 "Part (b):" }{TEXT -1 111 "  Here we look at the slope between t
he points (u,f(u)) and (x,f(x)) and then take the limit as u approache
s x." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "DQ := (f(u)-f(x))/(u-x);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Limit(DQ,u=x);\nvalue(%);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "Let's check the answer and r
elate it to the solution found in part (a).  All we need to do is to s
ubstitute " }{XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT 
-1 23 "  for x (and simplify):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "s
ubs(x=Pi/2,%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(
%);" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 
39 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }