Question

Evaluate the following limits, making sure to justify your calculations as much as possible:
as x approached 3-,3+,3 (|x-3|)/(3x^2-5x-12)

Answer 1

my mistake, I pressed submit too early but the question is still fully intact. I will write my steps so far in here. so at first I tried factoring to the point I got (|x-3|)/ (3x+4)(x-3) but then I got stuck since I dont think I can factor the absolute value out. if i subbed in the X then you get 0/12 which I dont think is the answer

Answer 2

|x-3|/(x-3) = 1 if x-3 > 0
|x-3|/(x-3) = -1 if x-3 < 0
so, from the left (x-3 < 0), you have the limit as
(|x-3|)/(3x^2-5x-12) = -1/(3x+4) = -1/13
from the right, it would be 1/13
the two-sided limit does not exist.