Can I solve $\lim_{(x,y)\to\ (0,0)} \frac{x^2y^2}{x^2+x^2y^2+y^2}$ by converting to polar coordinates?

You don't have to make it more complicated. Simply observe that $x^2 \le x^2+x^2y^2+y^2\implies \dfrac{x^2y^2}{x^2+x^2y^2+y^2} \le y^2\implies \text{limit} = 0$ .

Directly by polar we obtain

$$\frac{x^2y^2}{x^2+x^2y^2+y^2}=\frac{r^2\cos^2\theta \sin^2\theta}{1+ r^2\cos^2\theta \sin^2\theta}\to \frac{0}{1+0}=0$$