How to calculate $\lim_{(x, y) \to (0,0)} \frac{xy^2}{x^2 - y^2}$

Note that by $x=t+t^2$ and $y=t$ with $t\to 0^+$

$$\frac{xy^2}{x^2 - y^2}=\frac{t^3+t^4}{2t^3+t^4 }=\frac{1+t}{2+t } \to \frac12$$

therefore, since you also find paths with limit equal to zero, the limit doesn't exist.