Prove the following limit converges using an $\epsilon / \delta $ argument

Question

$\lim_{(x,y)\to(0.0)} \frac{x^2y}{x^2+y^2}$ this time i am reasonably sure the limit IS 0.

let $\epsilon >0 $ be given wts $\exists \delta >0 $ s.t $|\vec{x}-\vec{x_0}| < \delta \implies $ $|\frac{x^2y}{x^2+y^2} -0 |< \epsilon$ the bottom kind of looks like delta squared? im not bad at single variable continuity but multivariable is a terribly different ball game any hints much appreciated.

Answer 1

\begin{align*} \left|\dfrac{x^{2}y}{x^{2}+y^{2}}\right|&=\dfrac{x^{2}}{x^{2}+y^{2}}|y|\\ &\leq|y|\\ &\leq\sqrt{|x|^{2}+|y|^{2}}\\ &<\delta\\ &=\epsilon \end{align*} by choosing $\delta=\epsilon$.