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Lim f'(x)=0 implies lim f(x)/x = 0

  1. Oct 9, 2011 #1
    1. The problem statement, all variables and given/known data
    Given that the function [itex]f[/itex] is differentiable on the interval [itex](a,\infty)[/itex] and that [itex]\lim_{x\rightarrow\infty}f'(x)= 0[/itex]. Show that [itex]\lim_{x\rightarrow\infty}\frac{f(x)}{x}= 0[/itex].

    2. Relevant equations

    3. The attempt at a solution
    I have a pretty good intuition of why this is true: As f(x) approaches infinity, its derivative approaches 0. This means that, at some point, the function slope becomes less than that of x.
    This means that when x approaches infinity, x is dominant over f(x) and thus f(x)/x => 0.

    I can't come up with any strict mathematical proof though, so I would be very greatful if someone could help me with this.
  2. jcsd
  3. Oct 9, 2011 #2


    Staff: Mentor

    L' Hopital's Rule?
  4. Oct 9, 2011 #3
    L'Hopital's rule is unfortunately not allowed in this particular problem.
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