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Math Help - Prove left-hand limit against neg. infinity

  1. #1
    DBA
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    Prove left-hand limit against neg. infinity

    Hello, I need some help with this:

    Prove: lim x --> -1- 5/(x+1)^3 = neg. infinity

    I am not sure how to do that.
    a=-1

    For a limit against neg. infinity I know that
    if 0 < |x-a| < delta then f(x) < N

    and for a left-hand limit I know
    if a-delta < x < a then |f(x) - L| < epsilon


    So I have
    5/(x+1)^3 < N if |x+1| < delta
    and
    -1-delta < x < -1 if ???

    I do not know how to combine them.
    Thanks for any hints..
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  2. #2
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    Quote Originally Posted by DBA View Post
    Hello, I need some help with this:

    Prove: lim x --> -1- 5/(x+1)^3 = neg. infinity

    I am not sure how to do that.
    a=-1

    For a limit against neg. infinity I know that
    if 0 < |x-a| < delta then f(x) < N

    and for a left-hand limit I know
    if a-delta < x < a then |f(x) - L| < epsilon


    So I have
    5/(x+1)^3 < N if |x+1| < delta
    and
    -1-delta < x < -1 if ???

    I do not know how to combine them.
    Thanks for any hints..
    Did you try graphing the function.

    If you graph it, it's really quite obvious...
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  3. #3
    DBA
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    Hello, I did graph it, but for me nothing is obvious. I need to proof it without a graph and I have no clue how...
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  4. #4
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    Quote Originally Posted by DBA View Post
    Hello, I did graph it, but for me nothing is obvious. I need to proof it without a graph and I have no clue how...
    Look on your graph.

    What happens to y as you approach x = -1 from the left?


    If I remember correctly, \epsilon \delta proofs are only used when you are trying to find a whole limit, not a left or right-hand limit... It's been a while since I've done \epsilon \delta proofs though...
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  5. #5
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    If x< -1 then x+1< 0 so (x+ 1)^3< 0 and \frac{5}{(x+1)^3}< 0. As x approaches -1, the denominator approaches 0 while the numerator stays 5- ignoring the sign for the moment, that fraction gets larger and larger. Because it is negative, the fraction goes to -\infty.
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