Math Is Fun Forum

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Limits: 1^infinity vs. (1-infinitesimal)^infinity

Let n go to infinity, but be careful.
The cosine function can be as large
as 1, and we know what 1^infinity is.
but if the cosine function is just a tad
off of 1, then we assume the limit of
.999^infinity is zero, but that jump is
a discrete jump, not continuous.
Hence, when infinity is involved, either
non-continuous things can happen, or
maybe not, and we just don't understand
it yet.  Anyone??

Take the limit as n goes to infinity.
See the problem?

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igloo myrtilles fourmis

Dragonshade

Member

Registered: 2008-01-16

Posts: 147

Re: Limits: 1^infinity vs. (1-infinitesimal)^infinity

but the problem doesnt exist in this case, since cosx cant take 1, 1-cox cant be 0

TheDude

Member

Registered: 2007-10-23

Posts: 361

Re: Limits: 1^infinity vs. (1-infinitesimal)^infinity

The limit can only be found in relation to x, it is not a constant.  Clearly the limit does not exist when x = kpi, since we have a divide by 0 error in that case.  When x does not equal kpi the cos^{n+1}x term will drop, leaving you with

This can probably be simplified further, but I'm too lazy to deal with the absolute values, so I leave that to you.  What we're left with is this:

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luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: Limits: 1^infinity vs. (1-infinitesimal)^infinity

for set of integers, use \mathbb{Z}, similary for naturals \mathbb{N}, reals \mathbb{R}, complex \mathbb{C}, rational \mathbb{Q} etc.

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The End Of All Things To Come.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Limits: 1^infinity vs. (1-infinitesimal)^infinity

Hence, when infinity is involved, either
non-continuous things can happen, or
maybe not, and we just don't understand
it yet.  Anyone??

If you haven't heard of this yet, you just stumbled on to sequences of functions.  Here is a link:

Linky

I've only flipped through that, but it seems to be rather complete.  Unfortunately, it does take rather advanced knowledge of analysis to understand.

The simplest example of "weird stuff" is the following function:

Where the domain is [0,1].  Also, let:

I can prove any of the following facts if you wish:

1. f_n is continuously differentiable for any value n on [0,1]
2. f is continuous for any value x on [0, 1)
3. f is not continuous at 1.
4. f_n does not uniformly converge to f.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Limits: 1^infinity vs. (1-infinitesimal)^infinity

Thanks everyone and especially Ricky for explaining all of this.

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igloo myrtilles fourmis