How do you prove that lim_(x->1)1/(x-1) doesnot exist using limit definition?

How do you prove that ${\lim}_{x \to 1} \frac{1}{x - 1}$ doesnot exist using limit definition?

May 13, 2018

You approach the limit from both sides.

Explanation:

(1)
$x \to 1$ from above: we take $x = 1 + \epsilon$, where $\epsilon$ is getting smaller and smaller. The function will be:
$\frac{1}{\cancel{1} + \epsilon - \cancel{1}} = \frac{1}{\epsilon}$
As $\epsilon$ gets smaller the function gets larger, or:
${\lim}_{\epsilon \to 0} \frac{1}{\epsilon} = + \infty$

(2)
$x \to 1$ from below: we take $x = 1 - \epsilon$, where $\epsilon$ is getting smaller and smaller. The function will be:
$\frac{1}{\cancel{1} - \epsilon - \cancel{1}} = - \frac{1}{\epsilon}$
As $\epsilon$ gets smaller the function gets negatively larger, or:
${\lim}_{\epsilon \to 0} - \frac{1}{\epsilon} = - \infty$

This means there is not one limit.
graph{1/(x-1) [-12.66, 12.65, -6.33, 6.33]}