Finding $\lim_{(x,y)\to (0,0)} \left(\frac{x^2 y}{x^4 + y^2}\right)$

Question

I have big problem with the the limit of the following function: $$\lim_{(x,y)\to (0,0)} \left(\frac{x^2 y}{x^4 + y^2}\right)$$ I have tried to estimate lower and upper bound for Three-Series Theorem and to to convert it to polar coordinates and it didn't work well :)

Answer 1

If $y=x$ the limit becomes $\lim\limits_{x\to0}x^3/(x^4+x^2)=0$. If $y=x^2$ it becomes $\lim\limits_{x\to0}x^4/(2x^4)=1/2$. The limit then does not exist.