limits of 2 variables with trigonometric terms I'm trying to determine the following limit
$$ \lim_{(x,y) \to (0,0)} \frac{x^2-\sin(x^2y^2)+y^2}{x^2+\sin(x^2y^2)+y^2}$$
I tried to use polar coordinates and then Taylor and got
$$ \lim_{r \to 0} \frac{2-2 \cdot \cos^2(\theta) \cdot \sin^2(\theta) \cdot \cos(r^2\cos^2(\theta)\sin^2(\theta))}{2+2 \cdot \cos^2(\theta) \cdot \sin^2(\theta) \cdot \cos(r^2\cos^2(\theta)\sin^2(\theta))}$$
Which does not say anything I guess? How can I solve this type of questions?

We have that for $xy=0$ the limit is equal to $1$, for $xy\neq 0$ we have
$$ \frac{x^2-\sin(x^2y^2)+y^2}{x^2+\sin(x^2y^2)+y^2}=1-2\frac{\sin(x^2y^2)}{x^2y^2}\frac{x^2y^2}{x^2+\sin(x^2y^2)+y^2}\to 1$$
indeed $\frac{\sin(x^2y^2)}{x^2y^2}\to1$ and
$$0\le \left|\frac{x^2y^2}{x^2+\sin(x^2y^2)+y^2}\right|\le \frac{x^2y^2}{x^2+y^2}$$
with
$$\frac{x^2y^2}{x^2+y^2}\to 0$$
