Answer
The limit does not exist.
Work Step by Step
The limit
$$ \lim\limits_{(x,y) \to (0,0)}\frac{y^2}{x^2+y^2}$$
does not exist. If one considers the paths along the lines $ y=mx $, we get
$$\lim\limits_{(x,y) \to (0,0)}\frac{y^2}{x^2+y^2}=\frac{m^2}{1+m^2}$$
which depends on the slope $ m $ of the line and hence the limit does not exist.
