How to prove the limit $\lim\limits_{n\to\infty}n\sin\frac{2\pi}{n}\cos\frac{1}{n}$ doesn't exist? I am at a loss...
how do I prove that the limit
$\lim_{n\to\infty}(n\sin\frac{2\pi}{n}\cos\frac{1}{n})$
doesn't exist?

The limit does exist, one may just write, as $n \to \infty$,
$$
n\sin\frac{2\pi}{n}\cos\frac{1}{n}=(2\pi)\times \frac{\sin\frac{2\pi}{n}}{\frac{2\pi}{n}}\times\cos\frac{1}{n} \to 2\pi \times 1\times 1=2\pi.
$$
