Finding multivariable limits for the function $\frac{3x^2y}{x^2+y^2}$

Question

Could anyone help me with this

Find $$\lim_{(x,y) \to (0,0)} \frac{3x^2y}{x^2+y^2}$$ if this limit exists

I tried using the squeeze theorem, but I could not find a suitable expression for the squeeze theorem as I could not find any number that allow me to squeeze the limit in between and cancel the denominator

Answer 1

Polar coordinates: $$\frac{3x^2y}{x^2+y^2}=\frac{r^3\cos^2\theta\sin\theta}{r^2}=r\cos^2\theta\sin\theta$$ and as $r=\|(x,y)\|$...

Answer 2

$x^2\leq x^2+y^2$, then $0\leq\dfrac{x^2}{x^2+y^2}\leq 1$, now we obtain $\left|\dfrac{3x^2y}{x^2+y^2}\right|\leq 3|y|$, since $|y|\to 0$ we get $\dfrac{3x^2y}{x^2+y^2}\to 0$.