Answer
See explanation
Work Step by Step
We want to approach from different directions and get mismatching limits, thus proving the limit DNE (Does not exist).
If we approach from the $y=x$ line as $x$ goes to $0$ we end up with
$\lim\limits_{(x,y) \to (0,x)} \frac{y^2}{x^2+y^2}=\lim\limits_{(x,x) \to (0,0)} \frac{x^2}{x^2+x^2}=\frac{1}{2}$.
Approaching from the $x$ axis where $x$ goes to $0$ and $y=0$ we get a limit of $0/x^2=0$.
And since $0\ne1/2$ we can conclude the limit does not exist.
