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Math Help - limit question

  1. #1
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    limit question

    If f: [0, +infinity) --> R is differentiable and lim f'(x) = L (as x tends to +infinity) then lim [f(x)/x] = L (as x tends to +infinity).

    help, anyone?
    thanks
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Use L'Hopital rule.


    Fernando Revilla
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  3. #3
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    Can you think of another solution without using L'Hopital rule?
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by jefferson_lc View Post
    Can you think of another solution without using L'Hopital rule?

    Is it forbidden?


    Fernando Revilla
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  5. #5
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    yes
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  6. #6
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by jefferson_lc View Post
    yes

    Well, in that case a possibility is to translate the proof of the L'Hopital rule with g(x)=x . Nobody will dare to say you have used it. Take into account that:


    \dfrac{f(x)-f(0)}{x-0}=f'(\xi),\quad \xi \in (0,x)

    by the Lagrange's theorem




    Fernando Revilla
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  7. #7
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    Actually, can we even use L'Hopital's rule? We don't know that lim f(x) = +infinity (as x tends to +infinity).
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  8. #8
    MHF Contributor FernandoRevilla's Avatar
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    Use:

    \dfrac{f(x)}{x}-\dfrac{f(0)}{x}=f(\xi)

    and

    \displaystyle\lim_{\xi \to +\infty}f(\xi)=L


    Fernando Revilla

    P.S. Reading quickly at the OP, I "instinctively" saw \lim_{x \to +\infty}f(x)=+\infty .
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  9. #9
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    Please, correct if I'm wrong.
    What you're saying is that by the Mean Value Theorem we can find \xi \in (0,x) such that \dfrac{f(x)}{x}-\dfrac{f(0)}{x}=f'(\xi)\quad, for all x>0. Yet I still don't see how we can infer that \displaystyle\lim_{x \to +\infty}\dfrac{f(x)}{x}=L since x \to +\infty does not imply \xi \to +\infty. Thus we cannot use the fact that \displaystyle\lim_{\xi \to +\infty}f'(\xi)=L.
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  10. #10
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    Quote Originally Posted by jefferson_lc View Post
    Actually, can we even use L'Hopital's rule? We don't know that lim f(x) = +infinity (as x tends to +infinity).
    If the denominator goes to infinity at the limit point, there's no need for the numerator to go to infinity as well, even though in many textbooks it is written that it is necessary.
    Check the proof you have of L'hospital's theorem for the case \frac{\infty}{\infty}. Does it really use the fact that the numerator goes to infinity?
    As FernandoRevilla pointed, you can rewrite the proof of L'hospital's rule to this particular case if you are not allowed to directly use the theorem.
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  11. #11
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    You're absolutely right. Thanks!
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