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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "# LIMITS of FUNCTION
S \n#\n#\n# We will want to use plot3d." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 177 "with(plots):\n#\n# FIRST FUNCTION:\n#\n# This functi
on does not have a limit at the origin. The limit of f(x,y)\n# as we m
ove towards\n# the origin depends on the direction we use.\n#\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f:=x*y/(x^2+y^2);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"fG*(%\"xG\"\"\"%\"yGF',&*$F&\"\"#F'*$F(F
+F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot3d(f,x=-2..
2,y=-2..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 220 "# The funct
ion has a ridge along the line x=y. Every point on this line\n# has va
lue 0.5, except\n# for the origin. If we look at the contourplot, we s
ee that the\n# contours are very close to one\n# another near the orig
in." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "contourplot(f,x=-2..
2,y=-2..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 251 "#\n# SECOND
 FUNCTION:\n#\n#  This function has the limit of zero as we approach t
he origin from\n# any direction.\n# However, the function has a differ
ent limit if we look at the parabola\n#  y=x^2.\n# Again, the contours
 are tightly bunched near the origin." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "g:=x^2*y/(x^4+y^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"gG*(%\"xG\"\"#%\"yG\"\"\",&*$F&\"\"%F)*$F(F'F)!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot3d(g,x=-2..2,y=-2..2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "contourplot(g,x=-2..2,y=-2..
2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "#\n# THIRD FUNCTION
:\n#\n# This function does have a limit as (x,y) tends to (0,0).\n# No
tice that the contours don't show the same bunching as in the\n# previ
ous two examples." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "h:=2*x
^2*y/(x^2+y^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG,$*(%\"xG\"\"
#%\"yG\"\"\",&*$F'F(F**$F)F(F*!\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "plot3d(h,x=-2..2,y=-2..2);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 31 "contourplot(h,x=-2..2,y=-2..2);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 137 "#\n# Now look at a COMPOSITION of two co
ntinuous functions.\n# x^2+y^2 is continuous everywhere. tan(r) is con
tinuous if Pi/2 < r <\n# 3Pi/2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 24 "p:=tan(sqrt(x1^2+y1^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"pG-%$tanG6#*$,&*$%#x1G\"\"#\"\"\"*$%#y1GF,F-#F-F," }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 98 "# We need to use polar coordinates to plo
t this, because we want to\n# control the value of x^2+y^2" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "x1:=r*cos(t); y1:=r*sin(t);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G*&%\"rG\"\"\"-%$cosG6#%\"tGF'" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G*&%\"rG\"\"\"-%$sinG6#%\"tGF'" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot3d([x1,y1,p],r=Pi/2+0
.1..3*Pi/2-0.1,t=0..2*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}}{MARK "0 0 0" 22 }{VIEWOPTS 1 1 0 1 1 1803 }