Previous year on VJIMC competition we got T-shirts with following problem
Define $$f(\alpha):=\limsup\limits_{n\to\infty,\,n\in\mathbb{N}}(\sin(n))^{n^\alpha}.$$ Find $f(1)$ and $f(2)$.
I found some topics dealing with $$\lim_{n\to\infty}(\sin(n))^n,$$ but that wasn't much helpful (or I just overlooked important parts).
Any advice how to solve it? Thx.
Possible approach. Imagine you're dealing with angles randomly selected from $0$ to $\frac{\pi}{2}$. For any $x, 0 < x < 1$, there is a probability $\frac{\pi-2\arcsin x}{\pi}$ of the sine of that angle being in the interval $(1-x, 1)$. As $x \to 0$, that probability goes asymptotically to $\frac{\sqrt{8x}}{\pi}$. (This comes from the Taylor series for $\sin x$ around $x = \frac{\pi}{2}$.)
Heuristically, that means that when $\alpha = 1$, you're taking $1-x$ to the $\frac{\pi}{\sqrt{8x}}$ power or, setting $n = \frac1x$, taking $1-\frac1n$ to the $\frac{\pi\sqrt{n}}{\sqrt{8}}$ power. What happens to the result, as $n \to \infty$?
What if $\alpha = 2$? (This one's trickier; you might have to account for the sign of the result.)