How to find the limit of given function

Question

How would I find this limit?

$\lim_{n \to \infty}\ n\ \sin\ (2\pi en!)$, where $e$ is the exponential.

Answer 1

You write $$e = \sum_{k=0}^\infty \frac1{k!}.$$ Then, you remember that $\sin 2 \pi l = 0,$ for an integer $l.$ Now, notice that when you multiply $e$ by $n!$ that "kills" the first $n+1$ terms of the infinite series, as far as $\sin $ is concerned, and you are left with something roughly equal to $1/(n+1)$ (you should do the computation). So, the answer should be $2\pi,$ but again, don't take my word for it.