I tried (and still try) to prove that $\sin (\log n)$ doesn't have a limit at $\infty$. I know it is enough to show that a subsequence of $\log n$ approaches, modulo $2\pi$, arbitrarily close to 2 distinct values $\alpha, \beta$ (such that $\sin \alpha \neq \sin \beta$), which is a much weaker statement than "$\ \frac{\log n}{2\pi}$ is equidistributed modulo $1$" (which is even false...).
My questions are:
How do you prove my specific case?
Is there a general theory of limit of $f \circ g (n)$ where $f$ is a periodic continuous function and $g$ is continuous, (possibly monotone increasing) function diverging to $\infty$ at $\infty$?
What is a good source about equidistribution?
$\sin(\log(k^m)) = \sin(m\;\log(k))$ is a subsequence of the form $\sin(m\alpha)$ and may be easier to work with. The values of $\alpha$ where the sequence $\sin(m\alpha)$ is convergent have to be pretty rare.