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Math Help - LaGrange maybe...

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

    Prove that if f(x) differentiable on (a, \infty) and if lim_{x\to \infty} f'(x)=0 then: 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 f(x) differentiable on (a, \infty) and if lim_{x\to \infty} f'(x)=0 then: lim_{x\to \infty} \frac{f(x)}{x}=0
    Assume  \lim_{x\to\infty}f(x)=\pm\infty otherwise this problem is trivial.

    So as  x\to\infty ,  \frac{f(x)}{x}  will have the indeterminate form  \frac{\infty}{\infty} so we can use L'hopital's rule:  \lim_{x\to\infty}\frac{f(x)}{x} = \lim_{x\to\infty}f'(x)=0 .
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    Super solution from a super Member!
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