Does this limit exist finitely? Find the value of the following (if it exists):

$$\lim_{n\to\infty}n\sin(2\pi en!)$$

Does it exist? I think that it doesn't exist but I can't prove it. Please help me.

Hint: If we truncate the series for $e$ just after the term $\frac{1}{n!}$, the truncation error is 
$$\frac{1}{(n+1)!}+\frac{1}{(n+2)!}+\frac{1}{(n+3)!}+\cdots.$$
This is less than
$$\frac{1}{(n+1)!}\left(1+\frac{1}{n+2}+\frac{1}{(n+2)^2}+\cdots\right).$$
Thus for some integer $Q_n$, we have
$$Q_n+\frac{n!}{(n+1)!}\lt en!\lt Q_n+\frac{n!}{(n+1)!} \frac{1}{1-\frac{1}{n+2}}.$$
