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Proving an improper integral

  1. Mar 12, 2005 #1
    Proving an indeterminate form

    Prove for all positive integers n that [tex] \lim_{x\rightarrow 0}x({lnx})^n=0 [/tex]

    Thanks for any help.
     
    Last edited: Mar 12, 2005
  2. jcsd
  3. Mar 12, 2005 #2

    shmoe

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    That's not an integral.

    Do you know l'Hopital's rule? Combine it with induction.
     
  4. Mar 12, 2005 #3
    oh...i didn't think of induction
    i kept doing it by l'hopital's rule and it came out infinity/infinity all the time.
    Thanks for the advice
     
  5. Mar 12, 2005 #4

    dextercioby

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    It's futile to use L'Hôpital's rule (you can't get a reasonable expression for

    [tex] \frac{d^{k}(\ln x)^{n}}{dx^{k}} [/tex]

    )

    Do a substitution:

    [tex] x=e^{-v} [/tex]

    The result is immediate.It's like comparing exp & a finite polynomial.Since "n" is fixed,the factor [itex] (-1)^{n} [/itex] bears no relevance...

    Daniel.
     
  6. Mar 12, 2005 #5

    Hurkyl

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    L'Hôpital's rule + induction works fine for me... just like Jameson said.
     
  7. Mar 12, 2005 #6

    shmoe

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    and how do you know what happens in this comparison if you aren't familiar with it? enter l'Hôpital... :tongue2:
     
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