Motivated by a recent MSE [question](https://math.stackexchange.com/q/2071683/9464) about the sequence of function $\cos(n!\pi x)$, I have read related several related questions:


* [On the behaviour of $\sin(n!\pi x)$ when $x$ is irrational.](https://mathoverflow.net/questions/34369/on-the-behaviour-of-sinn-pi-x-when-x-is-irrational)
* [Is there a limit of cos (n!)?](https://math.stackexchange.com/q/8690/9464)


Is the existence of $\lim\_{n\to\infty}\cos(n!\pi x)$ for given arbitrary irrational $x$ still an open problem? Are there any known related references/results for this question?



Is the existence of $\lim_{n\to\infty}\cos(n!\pi x)$ for given arbitrary irrational $x$ an open problem?
Weyl proved in 1916 (see Satz 21 on Page 348 in [his paper](http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002266423)) that if a sequence of real numbers $\lambda\_1<\lambda\_2<\dots$ grows sufficiently rapidly (and $\lambda\_n=n!$ satisfies the precise constraint there), then for almost every real number $x$ (in the sense of Lebesgue measure), the sequence of fractional parts $\{\lambda\_n x\}$ is dense in $[0,1]$.


As a consequence, for almost every $x\in\mathbb{R}$, the limit of $\cos(n!\pi x)$ does not exist. This answers your first question. I don't know the answer to your second question, but I expect that the limit of $\cos(n!)$ does not exist either.


