Here is a problem from an old comprehensive exam that I am trying to solve
Problem: let $f:\mathbb{R}^{2} \to \mathbb{R}$ be a function defined as follows:
$f\left ( x,y \right )=\frac{\left ( x^{2}-y \right ).y^{2}}{x^{4}+y^{2}}$ if $\left ( x,y \right )\neq \left ( 0,0 \right )$
$f\left ( x,y \right )=0$ if $\left ( x,y \right )=\left ( 0,0 \right )$
The question is : Investigate the differentiability of $f$ at the point $\left ( 0,0 \right )$
Here is what I did so far: $\frac{\partial f}{\partial x}\left ( 0,0 \right )=0 $ and $\frac{\partial f}{\partial y}\left ( 0,0 \right )=-1 $
Now I applied the condition for differentiability for $f$ at the point $\left ( 0,0 \right )$:
$lim_{\left ( x,y \right ) \to \left ( 0,0 \right )}\frac{\left \| f\left ( x,y \right )-f\left ( 0,0 \right )-\bigtriangledown f\left ( 0,0 \right ).\left ( x,y \right ) \right \|}{\sqrt{x^{2}+y^{2}}}$ has to be $0$ if $f$ is differentiable at the desired point.
After simplifying the above limit, I got the following limit:
$lim_{\left ( x,y \right ) \to \left ( 0,0 \right ) }\frac{x^{2}y^{2}+x^{4}y}{\left ( x^{2}+y^{4} \right ).\sqrt{x^{2}+y^{2}}}$ . Here is where I am stuck. I cannot evaluate this limit. I tried everything, like evaluating the limit through different paths... but nothing seems to work out for me.
Any help please on how to finish my proof?