Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Science Advisor
    Homework Helper

    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


    User Avatar
    Science Advisor
    Homework Helper

    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...

  6. Mar 12, 2005 #5


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

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


    User Avatar
    Science Advisor
    Homework Helper

    and how do you know what happens in this comparison if you aren't familiar with it? enter l'Hôpital... :tongue2:
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook