Question

Question #270be

Answer 1

Rewrite using the definition of absolute value.

Explanation:

`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.