1. Not finding help here? Sign up for a free 30min 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 problem

  1. Nov 11, 2012 #1
    1. The problem statement, all variables and given/known data

    Evaluate [tex] lim_{n→∞} \frac{ (n+1)^{\frac{1}{n+1}} }{n^{\frac{1}{n}}} [/tex]

    2. Relevant equations



    3. The attempt at a solution
    This is actually a part of a series problem I am trying to solve using the ratio test.I can't seem to figure out this limit and L'Hopital's doesn't work. Any hints?

    BiP
     
  2. jcsd
  3. Nov 11, 2012 #2

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Note that if [itex]\lim_{n \rightarrow \infty} n^{1/n}[/itex] exists, let's call it [itex]L[/itex], then both the numerator and denominator converge to [itex]L[/itex], so the limit of the fraction is 1.

    Indeed, [itex]\lim_{n \rightarrow \infty} n^{1/n}[/itex] does exist, so focus on proving that fact.
     
  4. Nov 11, 2012 #3
    hey i think the limit is 1

    1/(n+1) → 0
    and
    1/n → 0

    if both powers will go → 0 then everything0 = 1


    then you get 1/1 = 1
     
  5. Nov 11, 2012 #4

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    The answer is right, but the argument is invalid. It's true that [itex]x^{1/n} \rightarrow 1[/itex] if [itex]x[/itex] is a fixed positive number, but here we have [itex]x[/itex] growing to infinity while the exponent shrinks to zero. It is not automatically true that the limit will be 1.

    Consider for example
    [tex]\lim_{n \rightarrow \infty} (n^n)^{1/n}[/tex]
    Surely this does not converge to 1, since [itex](n^n)^{1/n} = n[/itex].
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Limit problem
  1. Limit problem (Replies: 3)

  2. Limit Problems (Replies: 8)

  3. Limit problem (Replies: 4)

  4. Limit problem (Replies: 14)

  5. Limits problem (Replies: 4)

Loading...