TITLE: When $x$ goes to $0$ , what happens to $\sin\left(\frac{1}{x}\right) $ and $\cos\left(\frac{1}{x}\right)$?
QUESTION [8 upvotes]: When $x$ approaches $0$, do $\sin\frac{1}{x}$ and $\cos\frac{1}{x}$ converge or diverge?
How do you show this?

REPLY [2 votes]: Try by contradiction, suppose $\lim_{x\rightarrow 0}\sin{(\frac{1}{x})}=l, \:l\in \mathbb{R}$
therefore, $\forall \epsilon >0, \exists \delta>0$ such that if $| x - 0|<\delta$ the $| f(x)-l | < \epsilon$.
By archimedian property there exists $a_{1},a_{2}$ such that $0<a_{1},a_{2}< \delta$ and $a_{1}=\frac{1}{2k\pi\ + \frac{\pi}{2}}$ and $a_{2}=\frac{1}{2k\pi}$, $k\in \mathbb{N}$
$\Rightarrow | 1-l|<\epsilon$ and $ |l| <\epsilon$
Which concludes the proof.