Limit of sin(sin(sin(x)))

Question

I had an exam with the exercise $$ \lim _{x\to 0}\left(\frac{\sin(\sin(\sin(x)))}{x}\right) $$

but I needed to solve it without using L'hopital rule but I was not sure how to solve it, do you know how to clear that out?

I understand that $$ \lim _{x\to 0}\left(\frac{\sin(x)}{x}\right)=1 $$

but I don't get how to clear the inner sinnus.

Answer 1

$$ \frac{\sin(\sin(\sin(x)))}{x} = \frac{\sin(\sin(\sin(x)))}{\sin(\sin(x))} \frac{\sin(\sin(x))}{\sin(x)} \frac{\sin(x)}{x} $$

Each of the multiplicands converges to $1$. Hence, so is their product.