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.
On Dec 26, 2012, at 2:42 AM, kirby urner <email@example.com> wrote:
>> >> On Dec 16, 2012, at 11:22 PM, kirby urner >> <firstname.lastname@example.org> 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