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How to use mean vaule theorem here

  1. Jan 18, 2009 #1
    i got this question:
    there is a function f(x) which is differentiable on (a,+infinity)
    suppose lim [f(x)]/x =0 as x->+infinity
    prove that lim inf |f'(x)|=0 as x->+infinity ?

    does this expression lim inf f'(x)=0 has to be true
    if not
    present a disproving example

    ?

    i was present whis this solution but i didnt quite understand it.
    "First consider [itex]\lim_{m\to\infty}\frac{f(2m)}{m}[/itex], let [itex]2m=x[/itex] and this limit becomes [itex]2\lim_{x\to\infty}\frac{f(x)}{x}=0[/itex]. So [itex]\lim_{x\to\infty}\frac{f(2x)}{x}[/itex] exists and equals [itex]0[/itex]. So [itex]\lim_{x\to\infty}\frac{f(x)}{x}=0\implies \lim_{x\to\infty}\frac{f(2x)-f(x)}{x}=\lim_{x\to\infty}\frac{f(2x)-f(x)}{2x-x}=0[/itex]. So now consider the interval [itex][x,2x][/itex] and apply the MVT letting x approach infinity."

    mean value theorem says [itex]f'(c)=\lim_{x\to\infty}\frac{f(2x)-f(x)}{2x-x}=0[/itex]
    i dont know what the value of c??

    and it doesnt prove
    lim inf |f'(x)|=0 as x->+infinity

    ??
     
  2. jcsd
  3. Jan 18, 2009 #2

    Mark44

    Staff: Mentor

    Thanks for including your work in your post!!!!

    Regarding your question about the Mean Value Theorem:
    The theorem says that if f is continuous on [a, b], and differentiable on (a, b), then there exists some number c in (a, b) such that
    [tex]f'(c) = \frac{f(b) - f(a)}{b - a}[/tex]

    The MVT doesn't tell you the value of c or how to find it; it just says that such a number exists.

    I believe that what you're reading is saying that since f'(c) = 0 for some number c in [x, 2x], then this is also true in the limit as x approaches infinity.
     
  4. Jan 18, 2009 #3
    so how to prove that
    lim inf |f'(x)|=0 as x->+infinity
     
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