Answer
Limit does not exist
Work Step by Step
Consider our approach : $(x,y) \to (0,0)$ along $y=kx^2$
This implies that $\lim\limits_{x \to 0}\dfrac{mk^4}{x^4+k^2x^4}=\lim\limits_{x \to 0}\dfrac{mk^4}{x^4(1+k^2)}=\dfrac{k}{1+k^2}$
This shows that there are multiple limit values when the approach is different, so, the limit does not exist at the point (0,0) for the function $f(x,y)=\dfrac{x^2y}{x^4+y^2}$.
