1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Limit of (1+ln x)/x as x -> 0

  1. May 9, 2012 #1
    1. The problem statement, all variables and given/known data

    lim (1+ln x)/x = ?
    x->0

    2. Relevant equations



    3. The attempt at a solution

    lim 1/x + (ln x)/x
    x->0
    I know that (ln x)/x approaches -infinity faster than 1/x approaches infinity so the limit = -infinity, but how do I express this analytically?
     
  2. jcsd
  3. May 9, 2012 #2
    Ok, I got it. It's quite simple actually, duh.
     
  4. May 9, 2012 #3

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    That's great to hear.

    By the way, welcome to PF !
     
  5. May 9, 2012 #4

    sharks

    User Avatar
    Gold Member

    Yep, you just had to use L'Hopital's Rule and the answer is infinity.
     
  6. May 13, 2012 #5
    Thanks. :)
     
  7. May 13, 2012 #6

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    That's not an indeterminate form. The Hospital rule doesn't apply.
     
  8. May 13, 2012 #7

    sharks

    User Avatar
    Gold Member

    Hi vela

    Here is my understanding of the problem:
    [tex]\lim_{n\to 0} \frac{1+\ln x}{x}=\lim_{n\to 0} \frac{1}{x}+\lim_{n\to 0}\frac{\ln x}{x}[/tex]
    [tex]\lim_{n\to 0} \frac{1}{x}=∞[/tex]
    [tex]\lim_{n\to 0}\frac{\ln x}{x}=\lim_{n\to 0}\frac{1}{x}=∞[/tex]
    The limit above is evaluated by using L'Hopital's Rule.
     
  9. May 13, 2012 #8

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    That's not an indeterminate form. The Hospital rule doesn't apply.
     
  10. May 13, 2012 #9

    sharks

    User Avatar
    Gold Member

    Agreed, -∞/0 is not an indeterminate form. I kept assuming n approaches ∞ :redface:

    To evaluate:
    [tex]\lim_{n\to 0}\frac{\ln x}{x}=\lim_{n\to 0}\frac{1}{x}.\ln x[/tex]
    [tex]1/0=+∞[/tex]
    [tex]\ln 0=-∞[/tex]
    Hence, the product is -∞
     
    Last edited: May 13, 2012
  11. May 14, 2012 #10
    So, so wrong.
     
  12. May 15, 2012 #11
    You wrote for n -> 0, but I assume that was a typo :tongue: Division by 0 is not defined for real numbers, so cannot exactly call it equal to positive infinity. Only,

    [itex]\lim_{x\to 0^{+}} \frac{1}{x} = +\infty[/itex]

    [itex]\lim_{x\to 0^{-}} \frac{1}{x} = -\infty[/itex]

    Which means that the original limit will not be two sided.
     
  13. May 15, 2012 #12
    The original limit is only meaningful approaching from the positive side, since the natural logarithm isn't defined on the negative side. So from the point of view of this problem, it is safe to say that the limit of [itex]1 / x[/itex] is positive infinity (unless we want to agree that a limit is only meaningful if it is defined from both sides, in which case this thread is over).
     
  14. May 15, 2012 #13

    sharks

    User Avatar
    Gold Member

    Hi Infinitum and Steely Dan

    Thank you both for the clarification.
    [tex]\lim_{n\to 0^+} \frac{1+\ln x}{x}=\lim_{n\to 0^+} \frac{1}{x}+\lim_{n\to 0^+}\frac{1}{x}.\ln x=(+\infty)+(+\infty.-\infty)=(+\infty)+(-\infty)[/tex] which is undefined.
     
    Last edited: May 15, 2012
  15. May 15, 2012 #14
    This is not the way to go about it. If you know the properties of the natural logarithm, it's very clear that [itex]\text{ln}(x) + 1 \approx \text{ln}(x)[/itex] if [itex]x[/itex] is very small. So really this problem is just the limit of [itex]\text{ln}(x) / x[/itex] (which is clearly [itex]-\infty[/itex]).
     
  16. May 15, 2012 #15

    sharks

    User Avatar
    Gold Member

    I overlooked that part but it's clear that -∞+1≈-∞. Thanks again. :smile:
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Limit of (1+ln x)/x as x -> 0
  1. Derivative of 1 / ln x (Replies: 6)

  2. Converge of 1/ln(x) (Replies: 14)

  3. Integral of 1/ln(x) (Replies: 6)

  4. E^-ln(x) = 1/x? (Replies: 1)

Loading...