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Math Help - Absolute value limit? (should be easy)

  1. #1
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    Question 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.
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  2. #2
    MHF Contributor Calculus26's Avatar
    Joined
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    Florida
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    1,271
    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
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  3. #3
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    thank you, this makes sense to me now.
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