$\lim_{x\rightarrow 0} \frac{\sin(\sin x)}{x}$

Question

Can someone suggest how to solve this limit?

$$\lim_{x\to 0}\frac{\sin(\sin x)}{x}$$

If I substitute $y=\sin x$ then $\sin(\sin x)=\sin y$ while $x=\arcsin y$. Then the limit becomes $$\lim_{y\to 0}\frac{\sin y}{\arcsin y}$$

but this form is more complicated than the first one...

Answer 1

Hint multiply top and bottom by $\sin x$ and break into a product of two limits.