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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 41 "Definition of Limit at W
ork: An Animation" }{TEXT 256 0 "" }{TEXT -1 0 "" }}{PARA 257 "" 0 "" 
{TEXT -1 18 "by Dr. Coreen Mett" }}{PARA 258 "" 0 "" {TEXT -1 29 "with
 editing by Mr. Gary Ford" }}{PARA 259 "" 0 "" {TEXT -1 25 "(last modi
fied on 2/2/01)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "
with(plots):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(plottools):" }
}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been \+
redefined\n" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 40 "Define the functi
on (continuous at x=c)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "f
:=(-122*x^4+97*x+79*x^8+49*x^9+57*x^2-10)/(45*x^4+49*x^10+77*x^8+54);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*&,.*$)%\"xG\"\"%\"\"\"!$A\"
*&\"#(*F+F)F+F+*&\"#zF+)F)\"\")F+F+*&\"#\\F+)F)\"\"*F+F+*&\"#dF+)F)\"
\"#F+F+\"#5!\"\"F+,*F'\"#X*&F4F+)F)F;F+F+*&\"#xF+F1F+F+\"#aF+F<" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "a:=1; #user defines limit ta
rget on x-axis" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "L:=limit(f,x=a);\nevalf(L);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG#\"\"#\"\"$" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"+nmmmm!#5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 55 "evalf(L); subs(x=a,f); evalf(%); #compare limit to f(
a)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+nmmmm!#5" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6##\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+nm
mmm!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "W:=2; #Set window
 size" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"WG\"\"#" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 28 "plot(f,x=a-W..a+W,L-W..L+W);" }}{PARA 13 
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{MPLTEXT 1 0 57 "horizline:= plot(L,x=a-W..a+W,color=magenta,thickness
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rCS]f]*e%F-7$$\"34++]2%)38HFG$\"3'3=NFF'GiWF-7$$\"\"$F*$\"3HBMq&ez9L%F
--%'COLOURG6&%$RGBG$\"#5F)$F*F*Fdbl-F$6%7S7$F($\"3ImmmmmmmmF-7$F4Fibl7
$F>Fibl7$FIFibl7$FaoFibl7$F[pFibl7$F`pFibl7$FepFibl7$FjpFibl7$F_qFibl7
$FdqFibl7$FiqFibl7$F_rFibl7$FerFibl7$FjrFibl7$F_sFibl7$FdsFibl7$FisFib
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$F`alFibl7$FealFibl7$FjalFibl-F_bl6&Fabl$\"*++++\"!\")FdblF]fl-%*THICK
NESSG6#\"\"#-F$6%7$7$$\"\"\"F*$!+LLLL8!\"*7$Fhfl$\"+nmmmEF\\gl-F_bl6&F
ablFdblFdblF]fl-%*LINESTYLEG6#F[bl-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG
6$;F(Fjal;FjflF^gl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 23 "#pause to examine graph" }}}}{SECT 0 {PARA 3 "" 0 
"" {TEXT -1 8 "Animate:" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 33 "User d
efined values for animation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
40 "epsilon := 1; #starting value of epsilon" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(epsilonG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "mult:=.7; #multiplication factor for each step in eps
ilon" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%multG$\"\"(!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "N:=10; #number of iterations
 (smallest epsilon will be mult^N*epsilon)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"NG\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
28 "min_epsilon:=mult^N*epsilon;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
,min_epsilonG$\"*\\_Z#G!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
82 "left:=a-W; right:=a+W; bottom:=evalf(L-W); top:=evalf(L+W); #defin
e viewing window" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%leftG!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rightG\"\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'bottomG$!+LLLL8!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%$topG$\"+nmmmE!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "
spread:=50; k:='k':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'spreadG\"#]
" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for k from 1 to N do" 
}}{PARA 0 "" 0 "" {TEXT -1 9 "         " }{MPLTEXT 1 0 69 "s:=solve(ab
s(f-L)=epsilon); #method fails if \"RootOf\" solutions only " }}{PARA 
0 "" 0 "" {TEXT -1 9 "         " }{MPLTEXT 1 0 175 "Num_roots:=nops([s
]); \n   for i from 1 to Num_roots do \n     if type(s[i],float) then \+
 \n       spread := spread, abs(s[i]-a);\n     fi; \n   od: \n   delta
 := evalf(min(spread));" }{TEXT -1 8 "\n       " }{MPLTEXT 1 0 49 "pf:
= plot([L+epsilon,f,L-epsilon],x=left..right, " }}{PARA 0 "" 0 "" 
{TEXT -1 15 "               " }{MPLTEXT 1 0 24 "color = [red,blue,red]
):" }}{PARA 0 "" 0 "" {TEXT -1 7 "       " }{MPLTEXT 1 0 4 "p1:=" }
{TEXT -1 1 " " }{MPLTEXT 1 0 141 "inequal( \{ y<=L+epsilon, y>= L-epsi
lon\}, x=left..right, \n     y = bottom..top, optionsexcluded=(color=w
hite),optionsfeasible=(color=blue)):  " }}{PARA 0 "" 0 "" {TEXT -1 8 "
        " }{MPLTEXT 1 0 86 "p2:= inequal( \{ x>=a-delta, x<=a+delta\},
 x=a-delta..a+delta, \n       y = bottom..top, " }{TEXT -1 1 " " }
{MPLTEXT 1 0 71 "optionsfeasible=(color=yellow),\n       optionsexclud
ed=(color=white) ):" }}{PARA 0 "" 0 "" {TEXT -1 8 "        " }
{MPLTEXT 1 0 92 "p[k]:=plots[display](pf,p2,p1,horizline,\n       vert
line, view=[left..right,bottom..top] ): " }}{PARA 0 "" 0 "" {TEXT -1 
7 "       " }{MPLTEXT 1 0 23 "epsilon:=mult*epsilon: " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }{MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "plots[display](p[j]$j=2..N,insequence=true);" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 
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