Limit of a two variable function $\frac{x^2\sin^2{y}}{2x^2+y^2}$ I am trying to find the limit as $(x,y) \rightarrow (0,0)$ of $\frac{x^2\sin^2{y}}{2x^2+y^2}$
I tried this:
$\frac{x^2}{2x^2+y^2}<1/2$ so using the squeeze theorem $\frac{x^2}{2x^2+y^2}<1/2 \sin^2{y}$
and the limit of the right handside is 0, so the limit of the function must be 0. But wolfram alpha says the limit doesnt exist. Where am I going wrong?

Since $$2x^2+y^2 \geq 2\sqrt{2}\,|xy|$$
it follows that:
$$\left|\frac{x^2\sin^2 y}{2x^2+y^2}\right|\leq\left|xy\cdot\frac{xy}{2x^2+y^2}\right|\leq\frac{|xy|}{2\sqrt{2}}$$
so the limit is zero.
