Answer
The limit does not exist
Work Step by Step
Consider $f(x,y)=\dfrac{x^2y}{x^4+y^2}$
Let us consider the approach: $(x,y) \to (0,0)$ along $y=mx^2$
Then, we get $\lim\limits_{x \to 0}\dfrac{mx^4}{x^4+m^2x^4}=\dfrac{m}{1+m^2}$
This shows that there are multiple limit values and thus the limit does not exist at the point $(0,0)$ for the given function.