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Monotone conitnuous function - find limits

  1. Mar 17, 2008 #1

    daniel_i_l

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    Gold Member

    1. The problem statement, all variables and given/known data
    if f is monotonic and continues in R, and [tex]\int^{\infty}_{a} f(x)dx[/tex] converges then
    [tex]lim_{x \rightarrow \infty} xf(x) = 0 [/tex]


    2. Relevant equations



    3. The attempt at a solution
    I know that if xf(x) converges at all then it has to convergs to 0. But how do I know that it converges?
    Thanks.
     
  2. jcsd
  3. Mar 17, 2008 #2

    Dick

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    Homework Helper

    Can you show that for any positive value of k however small, that f(x)<k/x for all x>N for N sufficiently large? Hint: suppose it's not.
     
  4. Mar 18, 2008 #3

    daniel_i_l

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    Gold Member

    Thanks.
     
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