Question

How do you find the limit of `|x-7|/(x-7)` as x approaches 7?

Answer 1

The left and right limits disagree, so there is no limit as `x->7`

Explanation:

If `x > 7` then `abs(x-7)/(x-7) = (x-7)/(x-7) = 1`

If `x < 7` then `abs(x-7)/(x-7) = (-(x-7))/(x-7) = -1`

Hence:

`lim_(x->7^+) abs(x-7)/(x-7) = 1`

`lim_(x->7^-) abs(x-7)/(x-7) = -1`

Since the right and left limits are unequal, there is no two sided limit at `x=7`.

graph{abs(x-7)/(x-7) [4.24, 9.79, -1.266, 1.508]}