TITLE: Limit of $(\sin\circ\sin\circ\cdots\circ\sin)(x)$
QUESTION [10 upvotes]: I'm trying to find this limit:
$$\lim_{n \to \infty} \underbrace{\sin \sin \ldots \sin }_{\text{$n$ times}}x$$
Thank you

REPLY [7 votes]: Here is a different approach that doesn't lead as directly to a rigorous proof as Ross Millikan's, but is more concrete and shows the rate of convergence. Let the $n$th member of the sequence be given by the function $x(n)$. Using the Taylor series of the sine function, and approximating the discrete function $x$ by a continuous one, we have $dx/dn\approx -(1/6)x^3$. Separation of variables and integration gives $x\approx \pm\sqrt{3/n}$ for large $n$.