What is $\mathop {\lim }\limits_{n \to \infty } \frac{{\tan (n)}}{n}$? What is $\mathop {\lim }\limits_{n \to \infty } \frac{{\tan (n)}}{n}$?
 A: We will use two theorems, the first theorem is a result by Minkowski:

Given any $\theta \notin \mathbb{Q}$ and $\alpha \notin \mathbb{Z}$ such that
  $x - \theta y - \alpha = 0$ has no integer solutions.
  For any $\epsilon > 0$, there exists infinitely many pairs of integers $p,q$ such that 
  $$|q(p - \theta q - \alpha)| < \frac14\quad\text{ and }\quad |p - \theta q - \alpha| < \epsilon$$

Apply this to $\theta = \frac{1}{\pi}$ and $\alpha = -\frac12$, there are infinitely many pairs of non-zero $p, q$ such that
$$\left|q - \pi\left(p+\frac12\right)\right| < \frac{\pi}{4|q|}
$$
Using the fact
$$\left|\tan x\right| \le \frac{4}{\pi}|x|\quad\text{ for }\quad |x| < \frac{\pi}{4}$$
We find
$$\frac{1}{|\tan q|} = \left|\tan\left(q - \pi\left(p+\frac12\right)\right)\right| \le \frac{4}{\pi}\left(\frac{\pi}{4 |q|}\right) = \frac{1}{|q|}
\quad\implies\quad \left|\frac{\tan q}{q}\right| \ge 1$$
This implies there are infinitely many positive integer $n$ such that $\displaystyle\;\frac{|\tan n|}{n} > 1$.
The second theorem is Hurwitz's theorem. 

For any $\theta \notin \mathbb{Q}$, there are infinitely many pairs of relatively prime integers $p, q$ such that
  $$\left|\theta - \frac{p}{q}\right| \le \frac{1}{\sqrt{5}q^2}$$

Apply this to $\theta = \pi$, there are infinitely many pairs of relative prime $p, q$ such that
$$|\tan p| = |\tan(p - \pi q)| \le 
\frac{4}{\pi\sqrt{5}|q|}
\le \frac{4}{\sqrt{5}|p|-1}$$
This implies there is a subsequence of $\displaystyle\;\frac{\tan n}{n}$ converges to $0$ as $n \to \infty$.
Combine these two results, we can deduce $\displaystyle\;\lim_{n\to\infty} \frac{\tan n}{n}$ doesn't exist.
A: Hint: You need to show that there are infinitely many integer numbers close to multiples of $2\pi$ and infinitely many integer numbers close to $\pi/2+2\pi k$. Numbers close to multiples of $2\pi$ will have a limit of almost $0$ and the others will have a limit almost $\pm$ infinitely large. It is difficult to construct such sequences as $\pi$ has no obvious patterns. But I am sure that someone will give an elegant proof to it.
