Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

At lunch a coworker was talking about how to calculate, say, the 100th digit of pi using a square around the circle, then a pentagon, etc, basically you end up taking the limit of the circumference as the number of sides n goes to infinity.

So I tried working out the math, but I got stuck at proving:

$$\lim_{n \to \infty} 2n\tan\frac{\pi}{n} = 2 \pi$$

Any ideas how?

share|cite|improve this question
up vote 4 down vote accepted

Putting $n=\frac1h, h\to0$ as $n\to\infty$

$$\lim_{n \to \infty} 2n\cdot\tan\frac{\pi}{n}$$

$$=2\lim_{h\to0}\frac{\tan \pi h}h$$

$$=2\pi\lim_{h\to0}\frac{\sin \pi h}{\pi h}\cdot \frac1{\lim_{h\to0}\cos\pi h}$$

We know, $\lim_{x\to0}\frac{\sin x}x=1$ and $\lim_{x\to0}\cos x=1$

share|cite|improve this answer
Ahh makes sense thank you! – user1956609 Jul 10 '13 at 18:21
@user1956609, my pleasure. Conversion of the limit $0$ by proper replacement has made things easier me in many cases – lab bhattacharjee Jul 10 '13 at 18:23

From Taylor series we know that $$\tan x=_0x +o(x)$$ and if $y\to+\infty$ then $\frac{x}{y}\to 0$ hence we have $$\tan\left(\frac{x}{y}\right)y=_\infty\left(\frac{x}{y}+o\left(\frac{x}{y}\right)\right)y=_\infty x+o(1)$$ so we can conclude.

share|cite|improve this answer
Hope you're having a great day! – amWhy May 21 '14 at 11:43

As shown in this answer, $$ \lim_{x\to0}\frac{\tan(x)}{x}=1\tag{1} $$ For $x\ne0$, $(1)$ is equivalent to $$ \lim_{y\to\infty}\frac{\tan(x/y)}{x/y}=1\tag{2} $$ multiplying $(2)$ by $x$ yields $$ \lim_{y\to\infty}y\tan(x/y)=x\tag{3} $$ The case $x=0$ is verified trivially.

share|cite|improve this answer

Hint: Do you know that $\lim_{x\to 0} \frac{\sin x}{x} = 1$?

share|cite|improve this answer

$\lim_{n \to \infty} 2n(tan\frac{\pi}{n}) = \lim_{n \to \infty} 2\pi \frac{\tan \frac{\pi}{n}}{\frac{\pi}{n}}=\lim_{x\to 0}2\pi \frac{\tan x}{x}= 2\pi$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.