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Homework Help: Order by asymptotic growth rate

  1. Feb 18, 2012 #1


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    1. The problem statement, all variables and given/known data
    I try to order given functions and I am stuck with evaluating the following:

    [itex]f(n)= (n+1)![/itex] and [itex]g(n)=n^{logn}[/itex]

    2. Relevant equations
    [itex]\lim_{n\to\infty}\frac{f(n)}{g(n)} = 0 [/itex]

    then g(n) is faster growing.

    [itex]\lim_{n\to\infty}\frac{f(n)}{g(n)} = \infty[/itex]

    then f(n) is faster growing.

    3. The attempt at a solution
    I would guess that [itex]n^{logn}[/itex] is the faster growing function because it is exponential.
    Thus, I write
    [itex]\lim_{n\to\infty}\frac{(n+1)!}{n^{logn}}[/itex] and would expect the result to be zero.

    My problem is that I do not know hot to take the limit of a factorial function and I also have a problem with the n^logn.
    Can someone help me with this? Maybe I can write the functions in a different way?

    Thanks for any help.
  2. jcsd
  3. Feb 19, 2012 #2
    Edit: Nevermind. Now I'm curious though.

    [STRIKE]I might just be going out on a limb here, but I think that looking at it from this perspective might help:

    By the definition of logarithm,
    [itex]log_{10}n = x[/itex]
    [itex]n = 10^x[/itex]

    I hope it helps at least[/STRIKE]
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