Absolute value limit? (should be easy)

Heres the question and answer:

find the limit x -> -2 for the function:

|x+2|
_____

x+2

So I graphed it out and saw
a gap where x -> -2 and so I said the limit does not exist,

which is correct according to the textbook. But the

textbook solves the question in a way I'm not familiar with

could someone please explain whats happening here?


lim
t-> -2 from the left =

- (x + 2)
_______ = -1

x + 2


lim
t-> -2 from the right =

(x + 2)
______ = 1

x + 2



Algebraically it makes sense to me, however I thought that I had to use direct substitution to get the answer, in which case it makes no sense to me.

Also, why does the absolute value become replaced with a negative sign when coming from the left. Is this a general rule I can apply to these types of questions?

Normally I would look at other questions in the book like this one, but this is the only one from the questions with absolute value.

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

thank you, this makes sense to me now.