Limits of 2-Variable Functions

Graph of the function f(x,y) = x2y2/(x2 +y2), suggesting that
the limit of f(x,y) as (x,y) --> (0,0) exists.
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.
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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