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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 21 "Limits and Continuity" }}
{PARA 0 "" 0 "" {TEXT -1 307 "Limits of functions of several variables
 can be tricky. Usually it is much easier to show that the limit does \+
not exist than to prove the existance of it. In order to show that the
re is no limit we only need to show that there are two different paths
 towards (x0,y0) along which f(x,y) has different limits." }}{PARA 19 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "wit
h(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 8 "Example:" }{TEXT -1 
1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "g := (x,y) -> (x^2-
y^2)/(x^2+y^2); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 " If we approa
ch (0,0) along the x-axis i.e. when y=0," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "limit(g(x,0),x=0); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 45 " But if we approach (0,0) along the line y=x," }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 18 "limit(g(x,x),x=0);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 24 "limit(g(x,y),\{x=0,y=0\});" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plo
t3d(g(x,y),x=-4..4,y=-4..4,axes=BOXED,numpoints=1000);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 293 " Therefore there is no limit because no \+
matter how close we get to (0,0) we can always find points (of the for
m (x,0)) where the function takes the value 1 and we can also find poi
nts (of the form (x,x)) where the function takes the value 0. The valu
e of the function does not go to any limit." }}}{EXCHG {PARA 0 "" 0 "
" {TEXT 257 9 "Example: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 
"f:=(x,y)->(x^2+y^2)/((sqrt(x^2+y^2+1))-1);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 49 "limit((x^2+y^2)/((sqrt(x^2+y^2-1))-1),\{x=0,y=0\})
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot3d(f(x,y),x=-4..4,
y=-4..4,axes=BOXED,numpoints=1000);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 30 "Exercise: page 765, number 19:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "f:=(x,y)->(2*x^2+3*x*y+6*y^2)/(3*x^2+5*y^2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "limit(f(x,y),\{x=0,y=0\});" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot3d(f(x,y),x=-5..5,y=-
5..5,axes=BOXED);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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