Recall the definition of absolute value

|x| = x if x>0

= -x if x <0

For |x+2| = x+2 if x>-2

= -(x+2) if x < -2

For the lim as x -> -2 from the right( i.e x> -2)

|x+2|/(x+2) = (x+2)/(x+2) = 1

lim as x -> -2 from the left ( i.e x < -2)

|x+2|/(x+2) = -(x+2)/(x+2) = -1