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
>
>
>
>

```

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