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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 11 "Math 2210-1" }}{PARA 257 "" 0 "
" {TEXT -1 15 "October 6, 2004" }}{PARA 258 "" 0 "" {TEXT -1 18 "Visua
lizing limits" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 21 "In the case that f:  " }{XPPEDIT 18 0 "R^2;" "6#*$)%\"RG
\"\"#\"\"\"" }{TEXT -1 3 "-> " }{XPPEDIT 18 0 "R;" "6#%\"RG" }{TEXT 
-1 77 " ,  it is convenient to study limit and continuity questions us
ing the graph " }{XPPEDIT 18 0 "z = f(x,y);" "6#/%\"zG-%\"fG6$%\"xG%\"
yG" }{TEXT -1 170 " of f.  We will discuss several interesting limits \+
as (x,y)->(0,0).  Some of these are examples or homework problems from
 the text section 13.3, or modifications of them." }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 21 "restart:\nwith(plots):" }}}{PARA 0 "" 0 "" 
{TEXT -1 37 "1)  Find the limit as (x,y)->(0,0) of" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 23 "f:=(x,y)->sin(x^2+y^2);" }}}{PARA 0 "" 0 "" 
{TEXT -1 78 "Is your answer consistent with the graph of f?  Is f cont
inuous at the origin?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "pl
ot3d(f(x,y),\nx=-2..2,y=-2..2,scaling=constrained,\ncolor=white,\naxes
=boxed,\ntitle=`can you explain this graph?`);\n" }}}{PARA 0 "" 0 "" 
{TEXT -1 92 "Remark:  Maple plots the graph of a function of 2 variabl
es parametrically.  For example, in" }}{PARA 0 "" 0 "" {TEXT -1 275 "t
he picture above, it takes the square x=-2..2, y=-2..2 and cross-hatch
es it with a 20 by 20 grid (you can change this default option).  Then
 it sends (x,y) ---->(x,y,f(x,y)), and traces where the grid goes.  Yo
u can recover the domain grid by looking at the plot from above." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "2)  Can y
ou find the limit as (x,y)->(0,0) of" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "g:=(x,y)->sin(x^2+y^2)/(x^2+y^2);" }}}{PARA 0 "" 0 "
" {TEXT -1 112 "Is your answer consistent with the graph of g?  Could \+
you define g(0,0) to make g(x,y) continuous at the origin?" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "plot3d(g(x,y),\nx=-2..2,y=-2..2,sc
aling=constrained,\norientation=[42,75],\ncolor=white,\naxes=boxed,\nt
itle=`correct limit?`);" }}}{PARA 0 "" 0 "" {TEXT -1 93 "3)  Here is a
 more interesting limit we will try to analyze, perhaps using polar co
ordinates:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "h:=(x,y)->x*y/
(sqrt(x^2+y^2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "plot3d
(h(x,y),\nx=-.5..(.5),y=-.5..(.5),scaling=constrained,\norientation=[1
16,51],\ncolor=white,\naxes=boxed,\ntitle=`limit exists?`);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "4)  How about this one:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "k:=(x,y)->x*y/(x^2+y^2);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "plot3d(k(x,y),\nx=-.5..(.5
),y=-.5..(.5),scaling=constrained,\norientation=[116,51],\ncontours=20
,\naxes=boxed,\ntitle=`limit exists?`);" }}}{PARA 259 "" 1 "" {TEXT 
-1 134 "If we use polar coordinates we can parameterize this surface w
ith \"r\" and \"theta\" (we should work this parameterization out in c
lass):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 177 "plot3d([r*cos(the
ta),r*sin(theta),sin(2*theta)/2],\nr=0..1,theta=0..2*Pi,\ngrid=[30,30]
,axes=boxed,\nscaling=constrained,\ncolor=white,\norientation=[116,51]
,\ntitle=`another view`);\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 539 "5)  You might think from the previous two exam
ples that if you just check along radial lines you can always figure o
ut whether a limit as (x,y)->(0,0) exists, or that using polar coordin
ate will solve every such problem.  Although such reasoning often suff
ices,  here's a cool example that shows you would be wrong.  The first
 function we graph has value 1 when y<x^2 or y>2*x^2, and is zero for \+
x^2<y<2*x^2.  Every radial limit of f as you approach the origin (alon
g rays) is zero, and yet the limit of f(x,y) as (x,y)-->0 is not defin
ed!" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 122 "plot3d(Heaviside((y-2*x^2)*(y-x^2)),\nx=-1..1,y=-1..
1,grid=[40,40],\naxes=boxed,\ncontours=20,\ntitle=`parabolic narrows I
`);\n" }}}{PARA 0 "" 0 "" {TEXT -1 199 "The first parabolic narrows fu
nction is discontinuous along the parabolas y=x^2 and y=2*x^2.  We can
 create a slight slope where the cliffs were to make it continuous eve
rywhere except at the origin:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 148 "plot3d(-(1-abs(y/x^2-2))*\nHeaviside(3*x^2-y)*Heaviside(y-x^2),
\nx=-1..1,y=-1..1,grid=[40,40],\ncontours=20,\naxes=boxed,\ntitle=`par
abolic narrows II`);" }}}{PARA 0 "" 0 "" {TEXT -1 161 "You have a home
work problem like this example, but with a \"simpler\" formula, on whi
ch you should study behavior along parabolas to deduce a limit does no
t exist." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "34" 0 }{VIEWOPTS 1 1 0 
1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }