Here are graphs illustrating some examples of limits from the textbook, Section 14.2.
Please give the applets a few seconds to load. The resolution has been set higher (plot_points=200) so that we can see the behavior of the function close to the origin.
Variant of Example 1
$\lim_{(x,y) \to (0,0)} \frac{x^2}{x^2+y^2}$ does not exist.
Example 2
$\lim_{(x,y) \to (0,0)} \frac{xy}{x^2+y^2}$ does not exist.
Example 3
$\lim_{(x,y) \to (0,0)} \frac{x y^2}{x^2+y^4}$ does not exist.
Variant of Example 4
$\lim_{(x,y) \to (0,0)} \frac{x y^2}{x^2+y^2} = 0$.