In this case i'm struggling to show that the partial derivatives with respect to x are continuous. The answers always brush over how you determine it like it trivial so i think i'm missing something. I understand that that you only need to consider the case of x=0 and y=0 for the following example and show that the limit is equal to the partial derivative at (0,0) but i'm struggling to do this for the following example. So I was wondering if somebody would be able to demonstrate the proof to show that the x component of the partial derivative is continuous at (0,0)?
$$f(x,y)=\left\{\begin{array}{l} \frac{x^2y^2}{\sqrt{x^2+y^2}},\:\text{if $(x,y) \not= (0,0)$;}\\ 0,\:\text{if $(x,y)=(0,0)$;} \end{array}\right.$$
$$\begin{cases} \dfrac {\partial f}{\partial x}(x,y)= \dfrac{x^3y^2+2xy^4}{\sqrt[3]{x^2+y^2}}\\ \dfrac {\partial f}{\partial x}(0,0)=\lim \limits_{h\to 0}\left(\dfrac {f(h,0)}{h}\right)=0\end{cases}$$
Note: I have changed the example. This one should have both continuous partial derivatives with them both equal to 0 at (0,0).