# How do you show the limit does not exist lim_(x->6)(|x-6|)/(x-6)

Mar 31, 2018

See explanation.

#### Explanation:

First if we write the function without the absolute value we get:

## f(x)={(1;x>6),(-1;x<6):}

So if we calculate the left- and rightside limits we get:

Leftside limit:

${\lim}_{x \to {6}^{-}} \frac{| x - 6 |}{x - 6} = {\lim}_{x \to {6}^{-}} \left(- 1\right) = - 1$

Rightside limit:

${\lim}_{x \to {6}^{+}} \frac{| x - 6 |}{x - 6} = {\lim}_{x \to {6}^{+}} \left(1\right) = 1$

The leftside limit is not equal to the rightside limit, so the limit does not exist.