Re: Limit problem from analysis

*To*: mathgroup at smc.vnet.net*Subject*: [mg21523] Re: [mg21495] Limit problem from analysis*From*: Andrzej Kozlowski <andrzej at tuins.ac.jp>*Date*: Fri, 14 Jan 2000 02:43:38 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

Here is a very short proof, which I think satisifes your conditions ( depending of course on what one thinks should be contained in the first few chapters of an analysis book): To show that lim x^(1/x) as x goes to infinity is 1 is equivalent to showing that lim log(x^(1/x)) as x goes to infinity is 0. This in turn amounts to showing that log(x)/x goes to 0 as x goes to infinity. But this follows from the well known L'Hospital rule. I am sure that a more elementary proof can be given but it is likely to be rather longer. > From: world at writemaster.com.xxx To: mathgroup at smc.vnet.net > Organization: Concentric Internet Services > Date: Wed, 12 Jan 2000 08:35:39 -0500 (EST) > To: mathgroup at smc.vnet.net > Subject: [mg21523] [mg21495] Limit problem from analysis > > I am teaching myself analysis, and I'm stuck on a particular problem. > The problem is to prove that the limit as n goes to infinity of n to > the (1/n) power is 1. The only allowed tools are those of the first > few chapters of a basic analysis textbook. The hint given in the book > is to use the binomial theorem. > > You can find the work I've done at > http://writemaster.com/public/mathPostings/ > There are two documents there, one a Mathematica notebook and the > other an MS Word document. Any suggestions for how to proceed, or > even outright solutions, would be welcome. > > I am not taking any classes, and you will not be helping me with a > test or homework. > > Steve Oppenheimer > writer at writemaster.com > > > >