Define $f: \mathbb{R^2} \to \mathbb{R}, \ f(x,y)=\frac{x^2y}{x^2+y^2}$.
How to prove that all directional derivatives of $f$ in $(0,0)$ exist?
I tried with:
Let $v=(v_1,v_2) \in \mathbb{R^2}$ with $v_2 \neq 0$.
Then $\partial_vf(0,0)=\lim\limits_{h\to0}\frac{f(hv_1,hv_2)-f(0,0)}{h}=\lim\limits_{h\to0}\frac{h^2v_1v_2}{h(h^2v_1^2+h^2v_2^2)}=\lim\limits_{h\to0}\frac{v_1v_2}{h(v_1^2+v_2^2)}$.
Since the denominator is zero, how to continue?
A directional derivative is equivalent to exploit the directions $y = \lambda x$
substituting gives
$$ \frac{\lambda x}{1+\lambda^2} $$
so as we can observe along a line $y = \lambda x$
$$ \lim_{h\to 0}\frac{\frac{\lambda( 0+h)}{1+\lambda^2}-0}{h} = \frac{\lambda}{1+\lambda^2} $$