{VERSION 4 0 "SUN SPARC SOLARIS" "4.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 
0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "#sfb limits 5 feb 03
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "# the function x*y/(x^2
+y^2) as (x,y)->(0,0) [compare with (x^2-y^2)/(x^2+y^2)]" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f:=x*y/(x^2+y^2);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 30 "# switch to polar co-ordinates" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "g:=eval(f,\{x=r*cos(theta),y
=r*sin(theta)\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "plot3
d([r*cos(theta),r*sin(theta),g],r=0.01..1,theta=0..2*Pi,shading=zhue,t
itle=\"no limit at orgin\",axes=boxed);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 81 "# the function x*y/sqrt(x^2+y^2) as (x,y)->(0,0)[comp
are with(x^3-y^3)/(x^2+y^2)]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 21 "f:=x*y/sqrt(x^2+y^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
30 "# switch to polar co-ordinates" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "g:=eval(f,\{x=r*cos(theta),y=r*sin(theta)\});" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "plot3d([r*cos(theta),r*sin(
theta),g],r=0.01..1,theta=0..2*Pi,shading=zhue,title=\"limit zero at o
rgin\",axes=boxed);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 84 "# The function xy^2/(x^2+y^4) as (x,y)->(0,0) [one
 has to avoid y=-x^2 in the graph]" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "f:=x*y^2/(x^2+y^4);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "g:=eval(f,\{x=r*cos(theta),y=r*sin(theta)\});" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Limit(g,r=0)=limit(g,r=0);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "a:=plot3d([r*cos(theta),
r*sin(theta),g],r=0.01..1,theta=0..2*Pi,shading=zhue,title=\"limit at \+
orgin?\",axes=boxed):a;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "
with(plots);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "#restrict t
he surface to the curve y=x^2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 74 "b:=spacecurve([y^2,y,eval(f,x=y^2)],y=0.01..1,thickness=3,color=
black):#b;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "display(a,b);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "#the above function is \+
well behaved on line y=k*x but not the parabola" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 130 "a2:=plot3d([r*cos(theta),r*sin(theta),g],r=0.
01..1,theta=0..2*Pi,shading=zhue,title=\"limit at orgin?\",axes=boxed,
numpoints=10000):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "displa
y(a2,b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "26
" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }