Let $f : \mathbb{R}^{2} \to \mathbb{R}$ be given by
$$f (x,y) =\begin{cases} x^{2}y/(x^{6}+2y^{2}) &\text{ if }(x,y) \neq (0,0)\\ 0&\text{ if }(x,y) = (0,0) \end{cases}$$
Is this function differentiable at $(0,0)$?
I am trying to prove that it is differentiable by concluding the derivative is continous at $(0,0)$ that is showing the partial derivative of $x^{2}y/(x^{6}+2y^{2}) $ is $0$ at $(0,0)$
But how to prove that this limit exists using epsilon-delta? Could someone show the steps?
Many thanks!
The limit at $(0,0) $ doesn't exist. If $y=x^3$, you get $$ \frac {x^5}{x^6+2x^6 }=\frac {1}{3x}, $$ so the limit blows up,