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0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 
2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 256 
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mes" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }}
{SECT 0 {PARA 256 "" 0 "" {TEXT -1 9 "2D Limits" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 21 "restart;\nwith(plots):" }}}{SECT 1 {PARA 257 "
" 0 "" {TEXT -1 18 "General Parameters" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 125 "Set some options for all plots. You can change these to \+
your liking (or simply change the settings for each individual plot).
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "setoptions3d(labels=[`x`,`y`,`z
`], shading=zhue, axes=frame):" }}}}{SECT 1 {PARA 257 "" 0 "" {TEXT 
-1 16 "Plots and Limits" }}{PARA 0 "" 0 "" {TEXT -1 113 "In  this work
sheet we are interested in exploring Maple's capabilities for visualiz
ing the graph of the function " }{XPPEDIT 18 0 "z=f(x,y)" "6#/%\"zG-%
\"fG6$%\"xG%\"yG" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
68 "Define some function, plot it and try to compute its bivariate lim
it" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "f := (x,y) -> (x^2*y^3+x^3*y
^2-5)/(2-x*y);\nplot3d(f(x,y), x=-1..1, y=-1..1, style=patchnogrid);\n
Limit(f(x,y), \{x=0 ,y=0\}) = limit(f(x,y), \{x=0 ,y=0\});" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 113 "Here the graph suggests that the limit s
hould exist. However, Maple can't determine it. Only along special pat
hs." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 224 "f := (x,y) -> x*sin((x+y)/4
);\nplot3d(f(x,y), x=Pi/2..3*Pi/2, y=Pi/2..3*Pi/2, style=patchnogrid);
\nLimit(f(x,y), \{x=Pi ,y=Pi\}) = limit(f(x,y), \{x=Pi ,y=Pi\});\nLimi
t(Limit(f(x,y), x=Pi), y=Pi) = limit(limit(f(x,y), x=Pi), y=Pi);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "The graph indicates problems. Inde
ed, the limit is different along different paths." }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 272 "f := (x,y) -> (x^2-y^2)/(x^2+y^2);\nplot3d(f(x,y),
 x=-1..1, y=-1..1, style=patchnogrid);\nLimit(f(x,y), \{x=0 ,y=0\}) = \+
limit(f(x,y), \{x=0 ,y=0\});\nLimit(Limit(f(x,y), x=0), y=0) = limit(l
imit(f(x,y), x=0), y=0);\nLimit(Limit(f(x,y), y=0), x=0) = limit(limit
(f(x,y), y=0), x=0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 183 "Again, t
he graph indicates problems. Looking only at the limit along the coord
inate directions would indicate that the limit is 0. Also, along any l
ine through the origin of the form " }{XPPEDIT 18 0 "y=m*x" "6#/%\"yG*
&%\"mG\"\"\"%\"xGF'" }{TEXT -1 68 " the limit is zero. However, along \+
parabolas we have nonzero values." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
479 "f := (x,y) -> (x*y^2)/(x^2+y^4);\nplot3d(f(x,y), x=-1..1, y=-1..1
, style=patchnogrid);\nLimit(f(x,y), \{x=0 ,y=0\}) = limit(f(x,y), \{x
=0 ,y=0\});\nLimit(Limit(f(x,y), x=0), y=0) = limit(limit(f(x,y), x=0)
, y=0);\nLimit(Limit(f(x,y), y=0), x=0) = limit(limit(f(x,y), y=0), x=
0);\nLimit(Limit(f(x,y), y=m*x), x=0) = limit(limit(f(x,y), y=m*x), x=
0);\nLimit(Limit(f(x,y), x=y^2), y=0) = limit(limit(f(x,y), x=y^2), y=
0);\nLimit(Limit(f(x,y), x=-y^2), y=0) = limit(limit(f(x,y), x=-y^2), \+
y=0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "This function is radiall
y symmetric, so we can substitute " }{XPPEDIT 18 0 "r=sqrt(x^2+y^2)" "
6#/%\"rG-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF+F," }{TEXT -1 35 " to d
etermine the limit with Maple." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 349 "
f := (x,y) -> (x^2+y^2)/(sqrt(x^2+y^2+1)-1);\nplot3d(f(x,y), x=-1..1, \+
y=-1..1, style=patchnogrid);\nLimit(f(x,y), \{x=0 ,y=0\}) = limit(f(x,
y), \{x=0 ,y=0\});\nLimit(Limit(f(x,y), x=0), y=0) = limit(limit(f(x,y
), x=0), y=0);\nLimit(Limit(f(x,y), y=0), x=0) = limit(limit(f(x,y), y
=0), x=0);\nLimit(r^2/(sqrt(r^2+1)-1), r=0) = limit(r^2/(sqrt(r^2+1)-1
), r=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "0 \+
0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }