Prove that:
$$\lim_{x \to 0} \frac{1}{x}$$
Is non existent.
This is my attempt:
Assume The limit $= L$ Some real number $L$
By the definition of one sided limits we get from right and left respectively:
For $\delta > 0, \epsilon > 0$ $$x< \delta \implies \left| 1/x - L \right| < \epsilon$$
$$-x < \delta_2 \implies \left| 1/x - L \right| < \epsilon$$
Let $\delta' = \min(\delta_1, \delta_2)$
Let $\epsilon = 1$
$$x< \delta \implies \left| 1/x - L \right| < 1$$
$$-x < \delta_2 \implies \left| 1/x - L \right| < 1$$
$$0 < \delta_1 + \delta_2 \implies 2|1/x - L| < 2$$
what else should I do?