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Thread: how a sequence relates to a derivative?

  1. #1
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    how a sequence relates to a derivative?

    i'm having some trouble with the ideas behind this question.
    f is a continuous, defferentiable function with f(0)=0
    given $\displaystyle a_{n}=nf(\frac{1}{n})$ prove that $\displaystyle \lim_{x\to\infty}=f^\prime(0)$

    so i need a way to relate the value that some sequence converges to and the first derivative of a related function evaluated at 0.

    i'm having a really hard time with this question. effectively i've got nothing on it and its been a good 5-7 hours so i don't think i can solve it with what i know already.
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  2. #2
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    Re: how a sequence relates to a derivative?

    $\displaystyle \lim_{n\to\infty}a_n=\lim_{n\to\infty}\frac{f\left (\frac{1}{n} \right)}{\frac{1}{n}}$

    This is the indeterminate form 0/0, so apply L'H˘pital's rule to get the desired result.
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  3. #3
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    Re: how a sequence relates to a derivative?

    so the numerator would be $\displaystyle f^\prime(1/n)*(1/n)^\prime $ by the chain rule

    and the denominator $\displaystyle f(1/n)^\prime$

    so then i have $\displaystyle \lim_{x\to\infty} a_n = \lim_{x\to\infty} f^\prime(1/n)$

    and since 1/n goes to 0 then it becomes

    $\displaystyle \lim_{x\to\infty} a_n = \lim_{x\to\infty} f^\prime(0)$

    and i'm done?
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  4. #4
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    Re: how a sequence relates to a derivative?

    Your numerator is correct, but the denominator would simply be (1/n)', which cancels with this same factor in the numerator that results from the chain rule.

    After that, what you wrote is correct, except let n go to infinity rather than x.
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    Re: how a sequence relates to a derivative?

    kk thanks.
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    Re: how a sequence relates to a derivative?

    $\displaystyle f^\prime(1/n)*(1/n)^\prime $ by the chain rule

    and the denominator $\displaystyle (1/n)^\prime$

    so then i have $\displaystyle \lim_{n\to\infty} a_n = \lim_{n\to\infty} f^\prime(1/n)$

    and since 1/n goes to 0 then it becomes

    $\displaystyle \lim_{n\to\infty} a_n = f^\prime(0)$

    and i'm done?

    so then this? sorry im exhausted.
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  7. #7
    MHF Contributor MarkFL's Avatar
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    Re: how a sequence relates to a derivative?

    Yes, that looks good!
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