[Math] Prove that $\lim_{x\rightarrow 0}\frac{f(x^2)-f(0)}{x}=0$ if $f:\mathbb{R}\rightarrow\mathbb{R}$ is differentiable at $x=0$

Question

Let the function $f:\mathbb{R}\rightarrow\mathbb{R}$ be differentiable at $x=0$. Prove that $\lim_{x\rightarrow 0}\frac{f(x^2)-f(0)}{x}=0$.

The result is pretty obvious to me but I am having a difficult time arguing it precise enough for a proof. What I have so far is of course that since $f$ is differentiable;
$$f'(0)=\lim_{x\rightarrow 0}\frac{f(x)-f(0)}{x}$$
exists.

Any help would be greatly appreciated.

Answer 1

HINT:

$$\frac{f(x^2)-f(0)}{x}=\left(\frac{f(x^2)-f(0)}{x^2}\right)x$$