Question #270be

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Answer 1 · 2016-11-10T03:09:07

Rewrite using the definition of absolute value.

$abs(u) = {(u,"if",u >= 0),(-u,"if",u < 0) :}$

In this question, we have $u = x-4$ so $u >= 0$ when $x >= 4$ and $u < 0$ when $x < 4$ .

So, we have

$abs(x-4) = {(x-4,"if",x >= 4),(-(x-4),"if",x < 4) :}$

And, therefore

$abs(x-4)/(x-4) = {(1,"if",x >= 4),(-1,"if",x < 4) :}$

$lim_(xrarr4^+)abs(x-4)/(x-4) = 1$

$lim_(xrarr4^-)abs(x-4)/(x-4) = -1$

The limits from the left and right are not equal, so

$lim_(xrarr4)abs(x-4)/(x-4)$ does not exist.