Can I solve this multi-variable limit with polar coordinates: $\lim_{ (x,y)\to(0,0)} \frac{y^2\sin^2(x)}{x^4+y^4}$?

Question

$$\lim_{ (x,y)\to(0,0)} \frac{y^2\sin^2(x)}{x^4+y^4}$$

I know that I could let $x=0$ and solve the limit, and then the same for $y=x$. But I want to know if I could solve it only with polar coordinates.

Answer 1

If we want proceed by polar coordinates as a first step we can use

$$\frac{y^2\sin^2(x)}{x^4+y^4}=\frac{\sin^2(x)}{x^2}\frac{y^2x^2}{x^4+y^4}$$

with $\frac{\sin^2(x)}{x^2} \to 1$ and then use polar coordinates for the second term.

Answer 2

Using user's answer $\frac{x^2y^2}{x^4+y^4}= \frac{r^4 \sin^2a\cos^2a}{r^4(\sin^4a+\cos^4a)}= \frac{\sin^2a\cos^2a}{\sin^4a+\cos^4a}$, where angle $a$ can assume any value as $r\to 0$. Therefore the expression has no limit as both $x,y\to 0$.