Squeeze theorem 2 bounded functions $$\lim_{(x,y) \to (0,0)} \frac{x^2 \sin (y)}{x^2+y^2} $$
Squeeze Theorem:
$$g(x) \leq f(x) \leq h(x)$$
$$-1 \leq \sin(y) \leq 1$$
$$- \frac{x^2}{x^2+y^2}\leq \frac{x^2 \sin (y)}{x^2+y^2} \leq \frac{x^2}{x^2+y^2} $$
Also
$$ 0 \leq \frac{x^2}{x^2+y^2} \leq 1 $$
What should be my next step
How does this comes out to be zero

I would use polar coordinates and equivalence of functions:
$$\frac{x^2 \sin (y)}{x^2+y^2}=\frac{r^2\cos^2\theta\sin(r\sin\theta)}{r^2}\sim_{r\to 0}r\sin\theta\cos^2\theta.$$
