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.