Limits of 2-Variable Functions |
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Graph of the function f(x,y) = x2y2/(x2
+y2), suggesting that the limit of f(x,y) as (x,y) --> (0,0) exists. |
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Graph of the function y/(x2
+y2). The blue contours indicate what
happens as we approach (0,0) on this graph along the positive x- and
y-axes. |
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The left-hand image shows several lines and one cubic curve
(y = x3) approaching the origin. The right-hand image shows the
images of these curves on the graph of x3y/(x6 + y2). The cubic curve follows
a ridge of height 1/2, whereas the straight lines
eventually cross this ridge on their way to the origin. This explains why the limit of this function as we approach (0,0) fails to exist. |
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