Part 4: More with Limits Along Different Paths
There are functions f( x,y) for which the limit
along every straight line through a point ( p,q) exists, but
in which the limit itself does not exist. That is, in order for
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lim
( x,y) ® ( p,q)
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f( x,y) = L |
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it is necessary that the limit of f( x,y) along every
path through ( p,q) be equal to L.
EXAMPLE 5 Evaluate the limit
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lim
( x,y) ® ( 0,0)
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x2y
x4+y2
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| (5) |
along every line through the origin and also along the path y = x2.
Solution: Along the y-axis, we have x = 0, so that along the y-axis we have
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lim
( x,y) ® ( 0,0)
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x2y
x4+y2
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= |
lim
( 0,y) ® ( 0,0)
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02y
04+y2
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= 0 |
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Every non-vertical line through the origin is of the form y = kx where k
is a number, so that along a line y = kx we have
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lim
( x,y) ® ( 0,0)
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x2y
x4+y2
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= |
lim
( x,kx) ® ( 0,0)
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x2kx
x4+k2x2
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= |
lim
( x,kx) ® (0,0)
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kx
x2+k2
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= 0 |
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Thus, the limit is 0 along every linear path through the origin. However,
along the curve y = x2, we have
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lim
( x,y) ® ( 0,0)
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x2y
x4+y2
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= |
lim
( x,x2) ® ( 0,0)
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x2·x2
x4+( x2) 2
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= |
lim
(x,x2) ® ( 0,0)
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x4
2x4
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= |
1
2
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That is, the limit (5) does not exist, even though along
every straight line through the origin it has the same value. In the figure
below, see if you can identify what is significant about the curve y = x2.
