TITLE: $\lim_{n\rightarrow\infty} n\left|e^\frac{i2\pi}{n}-1\right|$
QUESTION [1 upvotes]: $$\lim_{n\to\infty}n\left|e^\frac{i2\pi}n-1\right|$$
I know that it must be $2\pi$ since it's just approximating the perimeter of the circle, but I can't do the limit algebraically.
I tried using the De Moivre formula: $e^{i\theta}=\cos(\theta)+i\sin(\theta)$, didn't work.
Any tips?
Edit: solved using @xpaul advice since it was the thing I was doing but I missed a 2 in the expansion of $(1-\cos(\frac{\pi}{n}))^2$.
Also solved it using Robert Israel hint because it was very clever.

REPLY [0 votes]: Alternatively:
$$\lim_{n\to\infty}n\left|e^\frac{i2\pi}n-1\right|=\left|\lim_{n\to\infty}\frac{e^\frac{i2\pi}n-1}{\frac1n}\right|=|2\pi i|=\sqrt{(2\pi)^2}=2\pi.$$
Note:
$$\lim_{x\to 0} \frac{e^{ax}-1}{x}=a.$$
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