TITLE: Disprove the limit $\lim_{x\to 0}\frac{1}{x}=5$ with epsilon-delta
QUESTION [9 upvotes]: I understand how to prove a limit such as $\lim_{x\to 3}x^2=9$. Now I was wondering, can one also use the epsilon-delta method to disprove a limit such as:
$$\lim_{x\to 0}\frac{1}{x}=5$$
If so, how?
Thanks!
edit: what would a formal proof look like?

REPLY [2 votes]: If $\lim_{x \to 0} \frac{1}{x} = 5$ then since $\lim_{x \to 0} x =0$ we have $\lim_{x \to 0} x  \cdot \frac{1}{x} = 1 = 0$ which is a contradiction.