How to determine whether this function is differentiable at a point?

The derivative at $0$ is given by the limit

$$\begin{align} f'(0)&=\lim_{h\to 0}\frac{f(h)-f(0)}{h}\\\\ &=\lim_{h\to 0}\frac{f(h)}{h} \end{align}$$

if this limit exists. If $h>0$, then

$$\begin{align} f'(0)&=\lim_{h\to 0^+}\frac{\frac{h}{1+h}}{h}\\\\ &=1 \end{align}$$

If $h<0$, then

$$\begin{align} f'(0)&=\lim_{h\to 0^-}\frac{h^2}{h}\\\\ &=0 \end{align}$$

The right-side and left-side limits are not equal. Therefore, the derivative at $0$ does not exist.

Tags:

Limits

Calculus

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