```Date: Dec 26, 2012 1:27 PM
Author: Robert Hansen
Subject: Re: A Point on Understanding

Let's look at it this way. Consider a pizza and you are going to slice this pizza into 1, 2, 3, 4, ... slices. For each n, where n = 1, 2, 3, ... the interior angle of each slice will be 360/n, correct? And if for each n we sum up the interior angles we will get 360, correct?.As n -> infinity, the interior angles approach 0. The limit of 360/n as n approaches infinity is 0. Yet, the sum of the interior angles is always 360, or stated another way, n * 360/n = 360, a constant. But you are missing a step when you stated your contradiction.You are asking why does limit(n) * limit(360/n) as n->infinity not equal the limit(n * 360/n) as n->infinity.This is because the limit(n) as n->infinity does not exist, and thus, you can't multiply the limits.You did show that limit(360/n) exists and is zero.You did show that limit(n * 360/n) exists and is 360.But your final statement actually involves limit(n), which you never showed existed, and indeed, doesn't exist.Bob Hansen On Dec 26, 2012, at 2:42 AM, kirby urner <kirby.urner@gmail.com> wrote:>> >> On Dec 16, 2012, at 11:22 PM, kirby urner>> <kirby.urner@gmail.com> wrote:>> >>> Descartes' proved that adding the angular deficits>> of all such vertexes, no matter their number, yields>> a constant number, 720 degrees.  Ergo Sigma (360 - v)>> over all N = 720.  This proves the limit at each>> vertex is never zero, as every vertex contributes>> some tiny "tax" or "tithe" to the invariant constant>> 720.  720/N > 0. |360 - v| > 0 even as N -> infinity.>> >>> >>> Contradiction?>> >> A contradiction would involve two statements, but>> here there is only one, that the deficit (360 - v)>> approaches zero as N increases without bound. That a>> function has a limit at a point doesn't mean that the>> function exists at that point. The limit of 1/x as x>> increases without bound is 0, but 0 isn't in the>> range of 1/x, nor is infinity in the domain, nor can>> it even be in any domain.>> >> Bob Hansen> > The two statements would be:> > (a)  the limit as N->infinity (where N is the number> of vertexes on the sphere), is 0 (where 0 is the diff > between 360 and the number of degrees surrounding> a vertex).  The sphere approaches a limit of perfect > flatness at each point.> > (b)  the sum of the non-zero difference at every vertex> adds to a constant 720> > Whether the function is defined at the limit doesn't > matter to the contradiction's being intelligible:  the > epsilon / delta approach suggests a real limit of 0, i.e.> I can always get within your epsilon with the right > delta, yet those vertexes, not just at the limit but all> along the way, have a computable non-zero difference, i.e> | 360 - v | can get as close to 0 as we like, but could> never really be 0, even at a limit, as the constant 720 > is always the sum of the differences from 0.> > It would seem as if one line of reasoning is saying "at> the limit, it's 0" while another is saying "zero could> never be the limit as each vertex contributes a positive> amount to a constant total and +n > 0."> > Kirby
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