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

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Answer 1 · 2018-03-31T18:12:11

See 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.