Solutions

We have

$$\begin{align*} f'(0)&=\lim_{h\to0}\frac{f(h)-f(0)}{h} \\ &=\lim_{h\to0}\frac{g(h)\sin(1/h)}{h} \\ &=\lim_{h\to0}\sin\left(\frac{1}{h}\right)\left(\frac{g(h)-g(0)}{h}\right) \\ &=0, \end{align*}$$

which follows from Exercise 5.21 and the fact that $g(0)=g'(0)=0$.