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Re: A Point on Understanding
Posted:
Dec 26, 2012 1:27 PM
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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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