A limit that involves two variables

It is equal to $0$ according to Wolfram alpha.

The domain of the function should exclude the $x$-axis, that is the domain $D = \{ (x,y) : y \neq 0\}$.

Let $\epsilon > 0$, we choose $\delta = \frac{\epsilon}2$, if $(x,y) \in D$ and $\sqrt{(x-0)^2+(y-0)^2} < \delta$

then

$$ \left|2x\sin^2 \left( \frac1y\right)-0\right| = \left|2x\sin^2 \left( \frac1y\right)\right|\leq 2|x| \leq 2 \delta < \epsilon.$$

$$|2x\sin^2(1/y)|\le 2|x|\cdot |1|\le 2|x|.$$

Since $2|x|\to 0$ as $x\to 0$, the limit is not undefined.