Question

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

Answer 1

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->6^-}(|x-6|)/(x-6)=lim_{x->6^-}(-1)=-1`

Rightside limit:

`lim_{x->6^+}(|x-6|)/(x-6)=lim_{x->6^+}(1)=1`

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