Calculating $\lim_{n\to\infty}\sqrt{n}\sin(\sin...(\sin(x)..)$

I will deal with the case when $x_0 \in (0,\pi)$ If $x_0 \in (0,\pi)$ and $x_{n+1} = \sin x_n $, for $ n \geq 0$ then $x_1 \in (0,1] \subseteq (0,\pi/2)$, and it is easy to see that from that point onwards, $0<x_{n+1}<x_{n}$ and hence $x_n$ converges to a fixed point of $\sin$ which has to be $0$.

We have $$ \dfrac{1}{\sin^2 x} - \dfrac{1}{x^2} = \dfrac{x-\sin x}{x^3} \times \dfrac{x}{\sin x} \times \left(\dfrac{x}{\sin x} + 1\right) \to \dfrac{1}{3}$$ as $x \to 0$.

This implies, putting $x = x_n$ $$ \dfrac{1}{x_{n+1}^2} - \dfrac{1}{x_n^2} \to \dfrac{1}{3}$$.

The Ceasaro mean of above, $$ \dfrac{1}{n}\sum_{i=0}^{n-1}\left(\dfrac{1}{x_{i+1}^2} - \dfrac{1}{x_i^2}\right) = \dfrac{1}{n}\left(\dfrac{1}{x^2_{n}} -\dfrac{1}{x^2_0}\right)$$ must also converge to $\dfrac{1}{3}$ and since $x_n > 0$, $ \sqrt{n} x_n \to \sqrt{3}$.

De Bruijn proves this asymptotic for the sine's iterates:

$$ \sin_n x \thicksim \sqrt{\frac{3}{n}} $$

Now we have:

$$ \lim_{n\to\infty} \sqrt{n} \sin_{n}{x} $$

We have $n\to\infty$.

$$ \lim_{n\to\infty} \sqrt{n} \sqrt{\frac{3}{n}} $$

$$ \lim_{n\to\infty} \sqrt{3} = \sqrt{3} $$

It is interesting to note that this result is independent of $x$. (As De Bruijn notes, G. Polya and G. Szegu prove a weaker result, namely, exactly this limit.)

This is only true for $x \in \left(0, \pi\right)$. For $x = 0$, the limit is $0$. For $x = \pi$, the limit is likewise, $0$.

For $\sin x$ negative, the limit goes to $-\sqrt{3}$. A proof follows. Note that the sine function is odd, that is:

$$ \sin_n (-x) = -\sin_n x$$

Now, we have:

$$ \lim_{n\to\infty} \sqrt{n} \sin_n (-x) $$

Or:

$$ -\lim_{n\to\infty} \sqrt{n} \sin_n (x) $$

Which we know to be $\sqrt{3}$, so:

$$ -\sqrt{3} $$

As a final summary ($k \in \mathbb{Z}$):

$$ \begin{cases} 0 & \mbox{if } x = k\pi \\ -\sqrt{3} & \mbox{if } x \in (2 \pi k - \pi, 2 \pi k) \\ \sqrt{3} & \mbox{if } x \in (2 \pi k, \pi + 2 \pi k) \\ \end{cases} $$