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Thread: LaGrange maybe...

  1. #1
    MHF Contributor Also sprach Zarathustra's Avatar
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    LaGrange maybe...

    Prove that if $\displaystyle f(x)$ differentiable on $\displaystyle (a, \infty)$ and if $\displaystyle lim_{x\to \infty} f'(x)=0$ then: $\displaystyle lim_{x\to \infty} \frac{f(x)}{x}=0$
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by Also sprach Zarathustra View Post
    Prove that if $\displaystyle f(x)$ differentiable on $\displaystyle (a, \infty)$ and if $\displaystyle lim_{x\to \infty} f'(x)=0$ then: $\displaystyle lim_{x\to \infty} \frac{f(x)}{x}=0$
    Assume $\displaystyle \lim_{x\to\infty}f(x)=\pm\infty $ otherwise this problem is trivial.

    So as $\displaystyle x\to\infty $, $\displaystyle \frac{f(x)}{x} $ will have the indeterminate form $\displaystyle \frac{\infty}{\infty} $ so we can use L'hopital's rule: $\displaystyle \lim_{x\to\infty}\frac{f(x)}{x} = \lim_{x\to\infty}f'(x)=0 $.
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    MHF Contributor Also sprach Zarathustra's Avatar
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    Super solution from a super Member!
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