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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }{TEXT 293 36 
"LIMITS OF FUNCTIONS OF TWO VARIABLES" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }
{TEXT 258 20 "Calculus III Project" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 259 10 "
Objective:" }}{PARA 3 "" 0 "" {TEXT 262 1 " " }{TEXT 263 51 "To illust
rate limits of functions of two variables." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "  Maple allows you to graph fun
ctions of two variables." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 
260 15 "Solved example:" }}{PARA 0 "" 0 "" {TEXT -1 22 "Consider the f
unction " }{TEXT 275 8 " f(x, y)" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "x^
2/(x^2+y^2);" "6#*&%\"xG\"\"#,&*$F$F%\"\"\"*$%\"yGF%F(!\"\"" }{TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "   a) Using calculus, find   " 
}{TEXT 276 12 "lim f(x,y)  " }{TEXT -1 2 "as" }{TEXT 292 9 "  (x,y)->
" }{TEXT -1 44 "(0,0) or show that the limit does not exist." }}{PARA 
0 "" 0 "" {TEXT -1 11 "   b) Plot " }{TEXT 277 7 "f(x,y) " }{TEXT -1 
145 "in an appropriate window to illustrate your finding in (a). Comme
nt on how the graph confirms (if it does) the result you obtained anal
ytically. " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "
" {TEXT -1 0 "" }{TEXT 261 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 
26 "   a) The limit along the " }{TEXT 278 1 "x" }{TEXT -1 7 "-axis (
" }{TEXT 279 2 "y=" }{TEXT -1 1 "0" }{TEXT 280 1 ")" }{TEXT -1 27 " is
 1; the limit along the " }{TEXT 281 1 "y" }{TEXT -1 7 "-axis (" }
{TEXT 282 1 "x" }{TEXT -1 58 "=0) is 0 : therefore the limit in questi
on does not exist." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 49 "   b) Plot the function in an appropriate window:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot3d(
x^2/(x^2+y^2),x=-1..1,y=-1..1,axes=boxed);\n" }}{PARA 13 "" 1 "" 
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Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 122 "It can be observed that in \+
fact the function is discontinous at (0,0) so it does not have a limit
 there. Moving along the " }{TEXT 283 1 "x" }{TEXT -1 58 "-axis on the
 surface means moving along the \"ridge\" where " }{TEXT 284 1 "z" }
{TEXT -1 34 " is one, whereas moving along the " }{TEXT 285 1 "y" }
{TEXT -1 75 "-axis on the surface means moving along the \"bottom\". T
here, the values of " }{TEXT 286 2 "z " }{TEXT -1 81 "are zero. ( You \+
can rotate the plot by clicking in the window with your mouse and" }}
{PARA 0 "" 0 "" {TEXT -1 26 " moving the bounding box.)" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{PARA 257 "" 0 "" {TEXT -1 0 "" }{TEXT 295 150 "
                                                                      \+
                                                                      \+
          " }}{PARA 258 "" 0 "" {TEXT 294 0 "" }{TEXT -1 0 "" }{TEXT 
257 10 "ASSIGNMENT" }{TEXT 265 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " \+
" }{TEXT 296 0 "" }{TEXT 271 9 "Problem 1" }{TEXT 297 1 ":" }{TEXT -1 
1 " " }}{PARA 3 "" 0 "" {TEXT 270 43 "  Find the indicated limits anal
ytically . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 1 " " }{TEXT 
-1 9 "   a) lim" }{XPPEDIT 18 0 "(x^3+y^3)/(x^2+y^2);" "6#*&,&*$%\"xG
\"\"$\"\"\"*$%\"yGF'F(F(,&*$F&\"\"#F(*$F*F-F(!\"\"" }{TEXT -1 8 "   as
  (" }{TEXT 287 4 "x,y)" }{TEXT -1 7 "->(0,0)" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 2 "  " }
{TEXT -1 8 "  b) lim" }{XPPEDIT 18 0 "2*x^2*y/(x^4+y^2);" "6#**\"\"#\"
\"\"*$%\"xGF$F%%\"yGF%,&*$F'\"\"%F%*$F(F$F%!\"\"" }{TEXT -1 8 "   as  \+
(" }{TEXT 288 3 "x,y" }{TEXT -1 8 ")->(0,0)" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 1 " " }{TEXT -1 9 "
   c) lim" }{XPPEDIT 18 0 "(x^2+y^2-2*x-2*y)/(x^2+y^2-2*x+2*y+2);" "6#
*&,**$%\"xG\"\"#\"\"\"*$%\"yGF'F(*&F'F(F&F(!\"\"*&F'F(F*F(F,F(,,*$F&F'
F(*$F*F'F(*&F'F(F&F(F,*&F'F(F*F(F(F'F(F," }{TEXT -1 8 "   as  (" }
{TEXT 289 3 "x,y" }{TEXT -1 9 ")->(1,-1)" }{TEXT 256 1 " " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 266 0 "" }
{TEXT 267 4 "    " }{TEXT 268 3 "d) " }{TEXT -1 3 "lim" }{XPPEDIT 18 
0 "(x^2+y^2)*ln(x^2+y^2);" "6#*&,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(F(-%#ln
G6#,&*$F&F'F(*$F*F'F(F(" }{TEXT -1 8 "   as  (" }{TEXT 290 4 "x,y)" }
{TEXT -1 7 "->(0,0)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }{TEXT 261 1 " " }{TEXT 269 2 "  " }{TEXT 274 2 "e)" }
{TEXT 273 1 " " }{TEXT -1 3 "lim" }{XPPEDIT 18 0 "sin(x^2+y^2)/(x^2+y^
2);" "6#*&-%$sinG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF*F+F+,&*$F)F*F+*$F-F*F+
!\"\"" }{TEXT -1 8 "   as  (" }{TEXT 291 3 "x,y" }{TEXT -1 8 ")->(0,0)
" }}}{PARA 0 "" 0 "" {TEXT -1 4 "    " }}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 0 "" }{TEXT 272 10 "Problem 2:" }}{PARA 0 "" 0 "" {TEXT -1 
183 "     Plot each of the five functions in problems 1a -- 1e individ
ually in an appropriate window.  In each case comment on how the graph
 confirmes the result you obtained analytically." }}}{PARA 0 "" 0 "" 
{TEXT -1 76 "_________________________________________________________
___________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 161 "MSIP Grant #P120A80089-98:  \"Three Urban Calculus Ref
orm programs: Adopting the Best\" 1998-2001                           \+
                                       " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{MARK "13 2" 10 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 
0 1 2 33 1 1 }