Two-variable limit question $\lim\limits_{(x,y)\rightarrow (0,0)} \dfrac{x^2y^2}{(x^2+y^4)\sqrt{x^2+y^2}}$
How to solve this two-variable limit? Thanks :D

Switching to polar coordinates:
$$
\lim_{r\to 0}\frac{r^4\cos^2\varphi\sin^2\varphi}{(r^2\cos^2\varphi+r^4\sin^4\varphi)r} = \lim_{r\to 0} \left[r\frac{\cos^2\varphi\sin^2\varphi}{\cos^2\varphi+r^2\sin^4\varphi}\right]
$$
then we have to be careful and see what happens as $\varphi$ changes its values: let us consider $\varphi=\pi /2 + k\pi$, then our quantity vanishes identically and the limit is zero; however if $\phi \ne \pi /2 + k\pi $, for $r>0$ the fraction is always bounded since it can be expressed as:
$$
\frac{\sin^2\varphi}{1+r^2\sin^2\varphi \tan^2\varphi},
$$
therefore the limits is again zero.
