FUNCTIONS OF SEVERAL VARIABLES
371
hence,
and
as (*,?)-»• (0,0)
41.54
41.55
41.56
41.57
Find
if it exists.
Let y = mx. Then
Hence, as (x, y)—»(0,0) along the line y =
mx, X
2 l(x
2 +y
2 )-+l/(l + m
2 ). Therefore,
does not exist, since 1/(1 4- m
2 ) is a nonconstant function of m.
Find
if it exists.
Let y = mx. Then
as (jt,.y)-»(0,0)
Hence,
does not exist.
Find
if it exists.
Let y = mx. Then
as (x, .y)-» (0,0)
Now let y = —xe". Then
By L'Hopital's rule,
Hence, the limit does not exist.
Find
if it exists.
Let y = mx. Then
as (jr,.y)-»(0,0)
This is also true when m—Q. In that case, y=0 and (x3 + y3)/(x2 + y) = x—*0]. However, let
(x, y)-»(0,0) along y = -xe. Then
By L'Hopital's rule,