Question

what is the limit of:
lim x --> 5 |x-5| / x-5

the answer is DNE (does not exist) but I'm not sure how to calculate the limit from the left and right side of the function

Answer 1

recall the definition of |x|:
|x| = x if x ≥ 0
|x| = -x if x < 0

so,
if x < 5 (limit from the left), |x-5| = -(x-5)
so |x-5|/(x-5) → -1

similarly, |x-5|/(x-5) → 1 if x ≥ 5 (limit from the right)

Answer 2

To calculate the limit from the left side, you would take the limit as x approaches 5 from the left, which would be -1. To calculate the limit from the right side, you would take the limit as x approaches 5 from the right, which would be 1. Since the two limits are not equal, the limit does not exist.