TITLE: Prove $\lim_{x\to0}\cos(\frac{1}{x})$ does not exist
QUESTION [7 upvotes]: Prove $\lim_{x\to0}\cos\left(\frac{1}{x}\right)$ does not exist using $\epsilon$-$\delta$ proof. 
I think my professor wants this to be done by letting $L$ be arbitrary. Then have two cases: $L > 0$ and $L < 1$. Then I would pick an epsilon, for all delta, and pick an $x$.
Any help would be appreciated.

REPLY [2 votes]: Hint $\bf\#1$:
Well, I guess the 'best' way is to use the "Sequential Criterion":

Let $A\subset \Bbb R$ and let $c$ be a $\color{red}{\text{cluster point}}$ of $A$. A function $f:A\to\Bbb R$ has a limit $L=\lim\limits_{x\to c}f(x)$ if for every $\color{blue}{\text{sequence }}\langle x_n\rangle$ that converges to $c$, $\langle f(x_n)\rangle$ converges to $L$.

If you have not studied it (yet), let me know, so I will adjust the answer accordingly.
Hint $\mathbf{\#2}$:
Plot (Computed by Wolfram Mathematica):