# Thread: Differentiable definition problem

1. ## Differentiable definition problem

Suppose that the function $f: \mathbb {R} \rightarrow \mathbb {R}$ is differentiable at $x_0 = 0$. Prove that $\lim _{x \rightarrow 0} \frac {f(x^2)-f(0)}{x} = 0$

How should I approach the $f(x^2)$ here?

2. Originally Posted by tttcomrader
Suppose that the function $f: \mathbb {R} \rightarrow \mathbb {R}$ is differentiable at $x_0 = 0$. Prove that $\lim _{x \rightarrow 0} \frac {f(x^2)-f(0)}{x} = 0$

How should I approach the $f(x^2)$ here?
\begin{aligned}\lim_{x\to{0}}\frac{f(x^2)-f(0)}{x}&=\lim_{x\to{0}}\frac{f(x^2)-f(0)}{x-0}\\
&=\left[f(x^2)\right]'|_{x=0}\\
&=2xf'\left(x^2\right)|_{x=0}\\
&=0
\end{aligned}