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.