Vector calculus - Prove a function is not differentiable at (0,0)

[physics1000]

$f\left(x\right)=\begin{cases}\sqrt{\left|xy\right|}sin\left(\frac{1}{xy}\right)&xy\ne 0\\ 0&xy=0\end{cases}$
I showed it partial derivatives exist at $(0,0)$, also it is continuous as $(0,0)$
but now I have to show if it differentiable or not at $(0,0)$.
According to answers it is not and they proved by showing the vector coordinates at $(1,1)$ does not have a limit.
But I dont want a proof like that, I tried using definition and got stuck...
I know my Linear transformation is basically 0. but still got in trouble of the definition

 

[physics1000]

Hi, is not needed anymore, at the end I managed to understand why they did vector coordinates, it is pratically trivial as I see it now
Thank thought :)