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Math Help - Infinitely differentiable function question

  1. #1
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    Infinitely differentiable function question

    Suppose f:R -> R is infinitely differentiable with f(0) = 0. Prove that all derivatives of f at 0 are 0 if and only if  \lim_{x \to 0} = \dfrac{f(x)}{x^n} = 0

    I get the general idea and can prove tex] \lim_{x \to 0} = \dfrac{f(x)}{x^n} = 0 \implies f^{(n)} = 0 [/tex] but can't seem to get the other direction of the proof.

    I suppose I have to find the limit definition of f^{(n)}(0) and then simplify it to get the required limit but I can't seem to get the definition.

    EDIT: Would it be ok to say
    f''(0) = \lim_{x \to 0} \dfrac{f'(x)}{x} = \lim_{x \to 0} \dfrac{\lim_{x \to 0} \dfrac{f(x)}{x}}{x} = \lim_{x \to 0} \dfrac{f(x)}{x^2}
    for example and then prove the formula for the nth derivative by induction
    Last edited by cana; May 21st 2011 at 01:28 PM.
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  2. #2
    Super Member girdav's Avatar
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    Quote Originally Posted by cana View Post
    Would it be ok to say
    f''(0) = \lim_{x \to 0} \dfrac{f'(x)}{x} = \lim_{x \to 0} \dfrac{\lim_{x \to 0} \dfrac{f(x)}{x}}{x} = \lim_{x \to 0} \dfrac{f(x)}{x^2}
    for example and then prove the formula for the nth derivative by induction
    No, because f'(x) =\lim_{h\to 0}\frac{f(x+h)-f(x)}h (you can't use the same letter).
    You can apply the Taylor's theorem.
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  3. #3
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    Surely using the other definition  f'(c) = \lim_{x \to c} \frac{f(x)-f(c)}{x-c} would make that fine?
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