Consider $f: \mathbb R^2 \rightarrow \mathbb R$ given by $$f(x\_1, x\_2) = \frac {x\_1^3 -x\_2^4} {x\_1^2 + x\_2^2}$$ if $(x\_1,x\_2) \neq (0,0)$ and $$f(x\_1,x\_2) = 0$$ if $(x\_1,x\_2) \neq (0,0)$.

I've proven that $$\lim\_{t \rightarrow 0} \frac 1 t (f(th)-f(0)) = \frac {h\_1^3} {h\_1^2+h\_2^2}$$

Now, I'm wondering why this allow me to conclude that $f$ is not differentiable at the point $0 \in \mathbb R^2$ ?