# How to prove the limit <munder> <mo form="prefix">lim <mrow class="MJX-TeXAtom-ORD">

How to prove the limit $\underset{n\to \mathrm{\infty }}{lim}n\mathrm{sin}\frac{2\pi }{n}\mathrm{cos}\frac{1}{n}$ doesn't exist?
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Dolharf1t1y
The limit does exist, one may just write, as $n\to \mathrm{\infty }$,
$n\mathrm{sin}\frac{2\pi }{n}\mathrm{cos}\frac{1}{n}=\left(2\pi \right)×\frac{\mathrm{sin}\frac{2\pi }{n}}{\frac{2\pi }{n}}×\mathrm{cos}\frac{1}{n}\to 2\pi ×1×1=2\pi .$