Results 1 to 3 of 3
Like Tree2Thanks
  • 2 Post By Plato

Math Help - Proof that lim x-> inf (x ^ (1/x)) = 1

  1. #1
    Member
    Joined
    Jun 2012
    From
    Georgia
    Posts
    164
    Thanks
    20

    Proof that lim x-> inf (x ^ (1/x)) = 1

    I started this proof by letting f(x) = (x ^ (1/x)) - 1 and attempting to show that that converges to zero. It's pretty straightforward to establish that a limit exists--x > e^1 => f(x) > 0 and f'(x) < 0, so f is bounded below and decreasing. Now what? Should I try lim inf, knowing that the existence of a finite limit implies that that limit equals the lim inf?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,381
    Thanks
    1474
    Awards
    1

    Re: Proof that lim x-> inf (x ^ (1/x)) = 1

    Quote Originally Posted by phys251 View Post
    I started this proof by letting f(x) = (x ^ (1/x)) - 1 and attempting to show that that converges to zero. It's pretty straightforward to establish that a limit exists--x > e^1 => f(x) > 0 and f'(x) < 0, so f is bounded below and decreasing. Now what? Should I try lim inf, knowing that the existence of a finite limit implies that that limit equals the lim inf?
    Notation: \exp(t)=e^t.

    So {\lim _{x \to \infty }}{x^{\frac{1}{x}}} = {\lim _{x \to \infty }}\exp \left( {\frac{{\ln (x)}}{x}} \right) = \exp \left( {{{\lim }_{x \to \infty }}\frac{{\ln (x)}}{x}} \right) = \exp \left( 0 \right) = 1
    Thanks from phys251 and topsquark
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jun 2012
    From
    Georgia
    Posts
    164
    Thanks
    20

    Re: Proof that lim x-> inf (x ^ (1/x)) = 1

    Oh wow. Take e^(ln(f(x)), and get the limit inside the exp(), triggering L'Hopital's rule. Thanks!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [Abstract Algebra] Anyone care to proof-read a proof?
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: December 4th 2012, 01:13 PM
  2. Replies: 5
    Last Post: October 19th 2010, 10:50 AM
  3. Replies: 0
    Last Post: June 29th 2010, 08:48 AM
  4. Proof with algebra, and proof by induction (problems)
    Posted in the Discrete Math Forum
    Replies: 8
    Last Post: June 8th 2008, 01:20 PM
  5. proof that the proof that .999_ = 1 is not a proof (version)
    Posted in the Advanced Applied Math Forum
    Replies: 4
    Last Post: April 14th 2008, 04:07 PM

Search Tags


/mathhelpforum @mathhelpforum