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Math Help - Real Analysis: Derivatives and Limits of Functions

  1. #1
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    Unhappy Real Analysis: Derivatives and Limits of Functions

    So I added a picture of the problem I am working on. I can't begin to prove this because I don't even understand why it is true. It seems so trivial but can someone explain this?Real Analysis: Derivatives and Limits of Functions-28.16.png
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  2. #2
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    Re: Real Analysis: Derivatives and Limits of Functions

    start with a form of the definition of a derivative at a point a

    $f'(a) = \displaystyle{\lim_{x \to a}}\dfrac{f(x)-f(a)}{x-a}$

    for the if direction assume there exists $\epsilon(x)$ that goes to 0 as $x \to a$ and show this leads to the above limit

    for the only if direction assume the derivative exists, i.e. the above limit exists, and show the statement leads to requiring the limit of $\epsilon(x)$ to be as stated.
    Thanks from rkushner83
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