Let $\displaystyle a_0=1, a_{n+1}=sin(a_n)$.
Does the constant $\displaystyle \alpha$ which $\displaystyle n^{\alpha}a_n$ has positive and finite limit exist?
Yes: you can show that $\displaystyle \lim_{n\to\infty}\frac 1{a_{n+1}^2}-\frac 1{a_n^2}$ is a positive constant (I found $\displaystyle \frac 13$, but it doesn't matter). Therefore, you can show that $\displaystyle \lim_{n\to\infty}\frac 1{na_n^2}$ is this constant, hence $\displaystyle a_n\sim \frac{\sqrt 3}{\sqrt n}$.