Question

Help!!! Determine whether this limit exists, justify your answer.

Answer 1

\[\lim_{(x,y) \rightarrow (0,0)}\frac{ (x^2-y^2)^2 }{ (x^2+y^2)^2 }\]

Answer 2

If the limit is to exist, it doesn't matter on the path we travel to reach the point, but we can find two values that dont match due to their path:

If we take the path x = 0 y -> 0
\[ \lim_{(0,y) \to (0,0)} \frac{((0)^2-y^2)^2}{((0)^2+y^2)^2} =1\] But now try the line y = x, our limite becomes:
\[\lim_{y \to 0} \frac{(y^2-y^2)^2}{(y^2+y^2)^2} =\lim_{y \to 0} \frac{(0)^2}{(2y^2)^2} =0 \]

So, depending on the path, we obtain different results. The limit does not exist.