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{SECT 0 {EXCHG {PARA 18 "" 0 "" {HYPERLNK 17 "Real Analysis with Maple
" 1 ":ICTCM9.mws" "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Limits of
 Functions" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "Limit of a Function
 at a Point" }}{PARA 0 "" 0 "" {TEXT -1 15 "Find the limit " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f := x -> 1/(sqrt(x^3+1));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "a := 1:\nL := value(subs(x=a
, f(x))):\nLimit(f(x), x=a) = L;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "plot(\{L,f(x)\}, x=0..2);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 182 "Digits := 4:\nX1 := evalf([`x`,seq(a-(10-i)/100, i
=1..9)]):\nX2 := evalf([`x`,seq(a+(10-i)/100, i=1..9)]):\nY1 := map(f,
X1):\nY2 := map(f,X2):\nlinalg[augment](X1,Y1,X2,Y2);\nDigits := 10:" 
}}}{PARA 0 "" 0 "" {TEXT -1 50 "Algebraic, numeric, and graphic eviden
ce suggests " }{XPPEDIT 18 0 "Limit(f(x),x=1)=sqrt(2)/2" "/-%&LimitG6$
-%\"fG6#%\"xG/F)\"\"\"*&-%%sqrtG6#\"\"#F+F0!\"\"" }{TEXT -1 1 "." }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 1 "\"" }{XPPEDIT 18 0 "epsilon" "I(ep
silonG6\"" }{TEXT -1 1 "­" }{XPPEDIT 18 0 "delta" "I&deltaG6\"" }
{TEXT -1 8 "\" Proofs" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 
18 0 "epsilon" "I(epsilonG6\"" }{TEXT -1 35 " > 0. Then we must show t
here is a " }{XPPEDIT 18 0 "delta" "I&deltaG6\"" }{TEXT -1 20 " > 0, s
uch that, if " }{XPPEDIT 18 0 "abs(x-a)" "-%$absG6#,&%\"xG\"\"\"%\"aG!
\"\"" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "delta" "I&deltaG6\"" }{TEXT 
-1 7 ", then " }{XPPEDIT 18 0 "abs(f(x)-L)" "-%$absG6#,&-%\"fG6#%\"xG
\"\"\"%\"LG!\"\"" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "epsilon" "I(epsilo
nG6\"" }{TEXT -1 13 ". So consider" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "f(x)-L;\nexpr_1 := normal(\");" }}}{PARA 0 "" 0 "" 
{TEXT -1 28 "³Rationalize the numerator.²" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 112 "expr_2 := \n -expand(numer(expr_1)*(2+sqrt(2)*sqrt(x
^3+1)))/      expand(denom(expr_1)*(2+sqrt(2)*sqrt(x^3+1))) ;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "expr_3 := factor(numer(expr_
2))/denom(expr_2);" }}}{PARA 0 "" 0 "" {TEXT -1 11 "Substitute " }
{XPPEDIT 18 0 "delta" "I&deltaG6\"" }{TEXT -1 5 " for " }{XPPEDIT 18 
0 "x-1" ",&%\"xG\"\"\"F$!\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 34 "expr_4 := subs(x-1=delta, expr_3);" }}}{PARA 0 "
" 0 "" {TEXT -1 12 "Assume that " }{XPPEDIT 18 0 "delta" "I&deltaG6\"
" }{TEXT -1 17 " < 1. Then, for  " }{XPPEDIT 18 0 "abs(x-1)" "-%$absG6
#,&%\"xG\"\"\"F'!\"\"" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "delta" "I&del
taG6\"" }{TEXT -1 18 ", it follows that " }{XPPEDIT 18 0 "0<x" "2\"\"!
%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 13 
" < 2. Thence," }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "expr_4 < s
ubs(x=2,numer(expr_4))/subs(x=0,denom(expr_4));" }}}{PARA 0 "" 0 "" 
{TEXT -1 7 "So that" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "ineq_
1 := abs(f(x)-L) < 7*delta/(2+2^(1/2));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "ineq_2 := rhs(ineq_1) < epsilon;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 34 "delta < rhs(ineq_2)*(2+sqrt(2))/7;" }}}
{PARA 0 "" 0 "" {TEXT -1 8 "Choosing" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "delta < min(rhs(\"),1);" }}}{PARA 0 "" 0 "" {TEXT -1 
4 "and " }{XPPEDIT 18 0 "abs(x-1)" "-%$absG6#,&%\"xG\"\"\"F'!\"\"" }
{TEXT -1 3 " < " }{XPPEDIT 18 0 "delta" "I&deltaG6\"" }{TEXT -1 17 " g
uarantees that " }{XPPEDIT 18 0 "abs(f(x)-L)<epsilon" "2-%$absG6#,&-%
\"fG6#%\"xG\"\"\"%\"LG!\"\"%(epsilonG" }{TEXT -1 13 " establishing" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Limit(f(x),x=a) = L;" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Additional Activities" }}{PARA 0 
"" 0 "" {TEXT -1 57 "Limits of piecewise defined functions can be inte
resting." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "f := x -> piecew
ise(x<1,a*x^2,3-x^3);\n'f(x)' = f(x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 91 "f := x -> piecewise(x<1,x+2,x<2,2-x,1-x^2);\n'f(x)' =
 f(x);\nplot(f(x),x=0..5, discont=true);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "f := x -> piecewise(x<>0, sin(x)/x,1):\n'f(x)' = f(x)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "f := x -> piecewise(x<
>0, cos(x)/x,a):\n'f(x)' = f(x);" }}}}}}{MARK "1" 0 }{VIEWOPTS 1 1 0 
3 2 1804 }