Prove the limit does not exist for: $\lim_{(x,y)→(0,0)}\frac{xy^2}{x^2+y^2 }$

Question

Prove the limit does not exist: $$\lim_{(x,y)→(0,0)}\frac{xy^2}{x^2+y^2 }$$

I am supposed to use $y=x$ and $y^2=x$ to prove that the limit doesn't exist using two paths. I cannot figure out how to correctly use them as I keep getting the same limit. Please help.

Answer 1

$$\left|\frac{xy^2}{x^2+y^2}\right|\leq |x|\underset{(x,y)\to 0}{\longrightarrow }0.$$

Answer 2

Aliter: Taking the polar form $x=r\cos\theta, \;y=r\sin\theta,$ we have $\lim_{(x,y)\rightarrow(0,0)}\frac{xy^2}{x^2+y^2}=\lim_{r\rightarrow 0}r\cos\theta\sin^2\theta=0$ for all real values of $\theta.$