Let $ \displaystyle f(x, y)=\frac{x^3-y^3}{x^2+y^2} $ be a multivariable function.
Examine if it is differentiable at $x=(0,0)$.

*I proved that the limit of the partial derivatives at $x=(0, 0)$ are $1, -1$ respectively. It is also easy to note that the partial derivatives there are not continuous. Hence to examine differentiability I have to check the limit:*

$$\lim\_{\left ( x, y \right )\rightarrow (0, 0)}\frac{f(x, y)-f(0, 0)-\frac{\partial }{\partial x}(0, 0)x-\frac{\partial }{\partial y}(0, 0)y}{\left \| x, y \right \|}$$

*I'm having some difficulty proving that this limit does not exist. P.S: I proved that the functin is continuous at $x=(0, 0)$ via sequences.*