Show for $z=f(x,y)=\sqrt[3]{xy}$ that $f$ is continuous and that the partial derivatives $\partial z/\partial x$ and $\partial z/\partial y$ exist at the origin but that the directional derivatives in all other directions do not exist.

I know that $\partial z/\partial x=\frac{y}{3(xy)^{2/3}}$. How to prove that $\partial z/\partial x$ exists at the origin (i.e. that $\lim\limits\_{(x,y)\rightarrow(0,0)}\partial z/\partial x$ exists).

Moreover, if $(\partial z/\partial x)\_{(0,0)}$ and $(\partial z/\partial y)\_{(0,0)}$ exists, how could the directional derivatives $$D\_{(\alpha)}f(x,y)=\frac{\partial f}{\partial x}\cos\alpha+\frac{\partial f}{\partial y}\sin\alpha$$ not exist?

That's exercise 5 of 1.5b