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!

Evaluating limit of floor function

  1. May 14, 2015 #1
    1. The problem statement, all variables and given/known data

    How $$ \lim_{x\to\infty} \frac{nlogx}{[x]} = 0 $$ ? Here n∈ ℕ
    Here [x] is greatest integer or floor function of x.
    2. Relevant equations
    [x] = x - {x} where {x} is fractional part of x.

    3. The attempt at a solution
    I know floor function is not differentiable.
    We are getting here ∞/∞ form but can't apply L hopital rule as denominator is not differentiable.
    How the limit evaluates to zero?
     
  2. jcsd
  3. May 14, 2015 #2
    use sandwich theorem.
    hint: start from ##?\le [x]\le ?##
     
  4. May 14, 2015 #3
    0≤ {x} < 1
    Now what?
     
  5. May 14, 2015 #4
    No. i am asking about [x] and not {x}.
     
  6. May 14, 2015 #5
    Okay,
    -∞≤[x] ≤ ∞ but [x] ≠ fractions
     
  7. May 14, 2015 #6
    No. Write it as ##f(x)\le [x]\le g(x)##. what is f and g?
     
  8. May 14, 2015 #7
    Why, what is f(x) and g(x) here?
     
  9. May 14, 2015 #8
    Okay. Use ##x-1\le [x]\le x##
     
  10. May 14, 2015 #9
    Okay, what next?
     
  11. May 14, 2015 #10
    try to transform that inequality and make it look like ##\frac{lnx}{[x]}## by taking reciprocal and multiplying by lnx. will the inequality change?
     
  12. May 14, 2015 #11
    Then
    1/x≤ 1/[x]≤ 1/(x-1)
    lnx/x ≤ lnx/[x]≤ lnx/(x-1)
    Now?
     
  13. May 14, 2015 #12
    Good. Now from sandwich theorem, if you have ##f(x)\le g(x)\le h(x)## and ##\lim_{x \to a}f(x)=L=\lim_{x \to a}h(x)##, then you can say that ##\lim_{x \to a}g(x) = L##.
     
  14. May 14, 2015 #13
    Oh, great lnx/x will be zero when x approach infinity same with h(x). So, g(x) evaluates to zero. Thanks.
     
  15. May 14, 2015 #14

    Mark44

    Staff: Mentor

    Let's be more precise. ln(x)/x never becomes zero, but gets arbitrarily close to zero as x grows large without bound.
    Also, here's the proper notation:
    ##\lfloor x \rfloor##
    \lfloor x \rfloor

    For the ceiling function, it is:
    ##\lceil x \rceil##
    \lceil x \rceil
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Evaluating limit of floor function
  1. Evaluating Functions (Replies: 6)

  2. Evaluating limit (Replies: 8)

Loading...