multivariable calculus - Conditions to exploit Polar coordinates in limits.

Evaluate, $$\lim_{(x,y)\rightarrow(0,0)}f(x,y)=\lim_{(x,y)\rightarrow(0,0)}\dfrac{2x^2y}{x^4+y^2}$$

When I used polar coordinates with $x=r\cos\theta, y=r\sin\theta$,

$$\lim_{r\rightarrow0}\dfrac{r\cos\theta\sin2\theta}{r^2\cos^4\theta+\sin^2\theta}=0$$

But when I use path $y=x^2$,

$$\lim_{(x,y)\rightarrow(0,0)}\dfrac{2x^4}{2x^4}=1$$

Also from path $x=0$ or $y=0$ both gives,
$$\lim_{(x,y)\rightarrow(0,0)}\dfrac{2x^2y}{x^4+y^2}=0$$

From path knowledge, I can say Limit does not exist.

Why this occurred that I got two different values of limits from Polar and the path makes me put a question that when to employ polar coordinates method to compute limits? When can I ascertain that it gives the correct value? Why is it giving out the value $0$ even when limit DNE?

Please help!