How can we solve this limit: $\lim _{x\to \infty \:}\left(\frac{\tan\left(x\right)}{x}\right)$?

Question

The limit was $\lim _{x\to \infty \:}\left(\frac{\tan\left(x\right)}{x}\right)$ I tried several ways to solved it, but I didn't find the answer. I think that it doesn't have a limit, but I want to see others' opinions. What do you think? Thanks in advance.

Answer 1

As I see from your comments, you are in the right way. However, you need to prove it mathematically.

HINT: Try to find a sequence $\{a_n\}$ increasing to infinity such that $\{f(a_n)\}$ diverges.

[As you show your work by editing the original post I will keep adding small hints if needed.]

Answer 2

The limit does not exist. Here is a way to prove it.

You have, for $n \in \mathbb{N}$, $$ \lim _{n\to \infty \:}\left(\frac{\tan\left(2n\pi\right)}{2n\pi}\right)=\lim _{n\to \infty \:}\left(\frac{0}{2n\pi}\right)=0. $$ You also have, for $n \in \mathbb{N}$, $$ \lim _{n\to \infty \:}\left(\frac{\tan\left((2n+1)\frac{\pi}2-\frac1n\right)}{(2n+1)\frac{\pi}2-\frac1n}\right)=\lim _{n\to \infty \:}\left(\frac{\cos(1/n)}{((2n+1)\frac{\pi}2-\frac1n)\sin(1/n)}\right)=\frac1{\pi}\neq 0. $$