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
$ 0\le \sin t\le t,\ 0\le t\le {\pi \over 2}.$ Thus
$${x^2+y^2-x^2y^2\over x^2+y^2+x^2y^2}\le h(x,y)\le 1\quad (*)$$
As $x^2y^2\le (x^2+y^2)^2$ we get
$${1-(x^2+y^2)\over 1+(x^2+y^2) }\le h(x,y)\le 1$$
Thus $\displaystyle\lim_{(x,y)\to (0,0)}h(x,y)=1.$
Remark We can as well apply the polar coordinates to LHS of $(*).$ Then
$${x^2+y^2-x^2y^2\over x^2+y^2+x^2y^2}={1-r^2\sin^2\theta\cos^2\theta\over 1+r^2\sin^2\theta\cos^2\theta}\to 1, \ r\to 0^+$$
