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
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.
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)