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Math Help - Multivariable Epsilon Delta Proof

  1. #1
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    Multivariable Epsilon Delta Proof

    Hi MHF,

    I'm looking for a little help with an Epsilon Delta proof. I'm taking a deferred exam 6 months after the course ended, so I really need help here.

    The Question: Let f be a function defined on a set S in R^n and suppose Q is a limit point of S. If \lim_{P \to Q} f(P) = 2, prove from first principles that \lim_{P \to Q} [1/f(P)] = 1/2

    My Attempt By definition, we know that |P-Q| < \delta  ==> |f(P) - 2| < \epsilon . Suppose  \epsilon ' = 1/\epsilon

    This is really all I have and understand and is probably wrong. I apologize in advance that I may have many questions in the following days as I'm slowly realize I barely understand anything from this course any more and I am truly trying but this is making no sense to me and I have no reference (Textbook was rented).

    Any help would be greatly Appreciated.
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by rtplol View Post
    Hi MHF,

    I'm looking for a little help with an Epsilon Delta proof. I'm taking a deferred exam 6 months after the course ended, so I really need help here.

    The Question: Let f be a function defined on a set S in R^n and suppose Q is a limit point of S. If \lim_{P \to Q} f(P) = 2, prove from first principles that \lim_{P \to Q} [1/f(P)] = 1/2

    My Attempt By definition, we know that |P-Q| < \delta  ==> |f(P) - 2| < \epsilon . Suppose  \epsilon ' = 1/\epsilon

    This is really all I have and understand and is probably wrong. I apologize in advance that I may have many questions in the following days as I'm slowly realize I barely understand anything from this course any more and I am truly trying but this is making no sense to me and I have no reference (Textbook was rented).

    Any help would be greatly Appreciated.
    First of all 1/epsilon is a very huge number!
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  3. #3
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    have you given any thought as to how one might re-write |1/f(P) - 1/2| in terms of something involving f(P) - 2 (which you know you can make "smaller than epsilon")?
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  4. #4
    MHF Contributor Also sprach Zarathustra's Avatar
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    And also...
    \lim_{P\to Q} f(P)=2 hence in small \delta neighborhood the function f(P) is bounded. ( 2-\epsilon<f(P)<2+\epsilon)
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