limit was Re: Limes

From: Robin Chapman (rjc_at_ivorynospamtower.freeserve.co.uk)
Date: 02/21/05 

Date: Mon, 21 Feb 2005 13:49:23 +0000

Peter wrote:

> I need help to prove
>
> lim x^(1 - cos(x))
> x -> 0
>
> I tried with x^(1 - cos(x)) = x / x^cos(x) and then l'Hospital rule but

AAAAARRRGGHHH!!! Not the dreaded hospital!

Take logarithms: you get
(1 - cos x)log x.

As x -> 0 (from above)
1 - cos x = x^2/2 + O(x^4).
So you have x^2 log x/2 + O(x^4 log x).

So ... what is lim_{x -> 0} x^2 log x?
Does lim_{x -> 0} x^4 log x = 0? 

-- 
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
  "Elegance is an algorithm"
    Iain M. Banks, _The Algebraist_

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