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On limits

by Organic
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matt grime
#37
Apr5-04, 05:18 AM
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Organic, how on earth can you expect people to know exactly what you mean just by posting one example without explanation. How can you not have realized yet that this is physicsforums.net not psychicforums.net. The confusion is entirely your causing by not explaining what you want. From one picture we are supposed to understand that you only want curves with THIS set of properties. Well, that curve in that link is also convex, should the curves only be convex? It only has one root if we carry it on in a naive smooth fashion, it cuts the x axis at positive x, must the curve always do this? The tangent has positive slope at all points, must this be true as well?
Organic
#38
Apr5-04, 05:26 AM
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MVT or Rolle's theorem plus a little thinking
Examples of it can be found here:http://www.ies.co.jp/math/java/calc/rolhei/rolhei.html

But this is not the case that I show here:
http://phys23p.sl.psu.edu/~mrg3/math...I/newtons.html

I gave this example, and by this I mean that I am talking only about this example.

If instaed you want to speack about another types of curevs, then you are talking to yourself, not to me.
matt grime
#39
Apr5-04, 05:37 AM
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Erm, thanks, Organic, I do know what the MVT and Rolle's theorem state, and, shockingly, I know how to prove them, amazing, someone might think I was a mathematician or something.

WHy didn't you say you only cared about that one example.
Organic
#40
Apr5-04, 05:54 AM
P: 1,210
If I am not mistake we are in a thread that dealing with the limit problem, where there are infinitely many steps that cannot reach the limit point.

I gave this N-R example as something which is another example that is different from the epsilon-deltha method.

Is it not understood?
matt grime
#41
Apr5-04, 06:31 AM
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You are making a mistake. This thread didn't start about that. You hijacked it so that you could talk about your interests again. And I sadly came along too.

Different from? As you've not proven anything to do with convergence that's a little rich.
Organic
#42
Apr5-04, 07:48 AM
P: 1,210
If you look at the previous thread will see the you started this:
But is still makes no sense. x/0 is not a well-defined symbol in the real number system that one can manipulate like this.
matt grime
#43
Apr5-04, 08:01 AM
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And what's that got to do with Newton-Raphson iteration? We explained what the more formal interpretation of your beloved Wolfram definition of infinity is.
Organic
#44
Apr5-04, 08:10 AM
P: 1,210
x/0 deeply connected to the limit problem.
HallsofIvy
#45
Apr7-04, 06:21 AM
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Quote Quote by Organic
x/0 deeply connected to the limit problem.
Only in the minds of those who do not understand what a limit is.

"x/0" is "undefined" if x is a non-zero constant and "undetermined" if x is zero.

If you mean that x is a variable, then "x/0" makes no sense at all.

If you mean "lim as a-> 0 of x/a" then you should say that: the whole point of the theory of limits is that "x/0" will tell you nothing about the limit.
Organic
#46
Apr7-04, 07:00 AM
P: 1,210
1/0 is the same as 1*oo and in both cases we are no longer in a finite system.

The whole idea of the interesting point of view of the limit concept is that no infinitely many elements can reach the limit itself.

This unclosed gap which is > 0 cannot be closed by infinitely many elements.

therefore the sum of .999... or the intervals of N-R is undefined by definition.

For example, Cauchy method only forcing the impossible to be possible by "raping" infinitely many ... to have a sum.
matt grime
#47
Apr7-04, 07:11 AM
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You're wrong and veering off topic again with you own personal incorrect view of mathematics. Please stay on topic.

Counter examples: let x_n=0 if n is even 1/n n odd. this sequence converges to zero, adn reaches 0 infinitely often. OR let x_n^{M} be the sequence define to be 1 for n<M 0 other wise - this sequence converges to zero and is zero for all n>M. Demonstrate a non-zero real number between 0.9999.. and 1. Hint: can't be done. The infinite sum os defined. It is the limit of the partial sums. (N-R, or Newton Raphson, has no need to be here). I presume Chausy is Cauchy. I don't think you understand enough of the mathematics to be able to form an opinion about completions wrt norms. So, this is mathematics, in the real numbers in decimal notation 0.9999.. is the same as 1. It has been proven many times. If you're going to tell us we're wrong then please don't do so in this thread. Start another one and attempt to understand the answers that will be given. Don't hijack this one please - I've answered your post and told you where you're wrong conceptually as well as physically. If you don't accept that then you aren't using the mathematics correctly and you aren't adding to this thread's worth. Start one in TD say, but this topic has been done to death and that you cannot accept the PROOF is a reflection on you not the mathematics.
Organic
#48
Apr7-04, 07:16 AM
P: 1,210
Another example:

PI exact plece in the real line is unknown, because in any representation method of it we have to use infinitely many elements to define it.
Demonstrate a non-zero real number between 0.9999.. and 1
Demonstrate a zero gap between 0.999... and 1
matt grime
#49
Apr7-04, 07:39 AM
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Again? There are at least 2 proofs of this fact in this thread alone. Let x_n be the n'th partial sum 0f 0.9+.009+.0009...

|1-x_n|= 1/10^n

0.999.. =lim x_n

hence |1-0.999...| =0 as the difference with the limit tends to zero, ie can be made of arbitrarily small absolute value.

If you disagree with that then you are disagreeing with the definition of the real numbers. Got it? If you want to work in a different number system then start a different thread or something.

Just realized this isn't in the thread I thought it was in (new page carry error, bane of mathematics) so rant away in your own private langauge at will.
Zurtex
#50
Apr7-04, 07:48 AM
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Quote Quote by Organic
Another example:

PI exact plece in the real line is unknown, because in any representation method of it we have to use infinitely many elements to define it.

Demonstrate a zero gap between 0.999... and 1
How do you mean it is unknown?

I'm fairly sure it is at [itex]\pi[/itex]... If you let [itex]\pi[/itex] be your base unit then it is really easy to mark it on.

Or do you just mean there is no given ratio between 1 and [itex]\pi[/itex] in terms of decimals?
matt grime
#51
Apr7-04, 08:00 AM
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Here's a little thing you need that you don't seem to know, Organic.

Suppose a and b are real numbers and for any e>0 we know |a-b|<e then a=b.

proof: if a is not b then a=b is nonzero. let d be the difference let e = d/2 then |a-b|=d and |a-b| <d/2, contradiction, hence d is zero.
Organic
#52
Apr7-04, 09:10 AM
P: 1,210
Suppose a and b are different real numbers and for any e>0 we know |a-b|<e then a not= b.

proof:

If a is not b then |a-b|>0.

Let d be the difference.

Let e = d/2 then |a-b|=d and |a-b| < d/2 > 0, hence d > 0
and also |a-b|/2 > 0.

therefore non-zero/2 > 0.
matt grime
#53
Apr7-04, 09:15 AM
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But your hypothesis is false: if a and b are distinct real numbers then it is not true that for every e>0 |a-b|<e. You do understand what the quantifier for all means?
matt grime
#54
Apr7-04, 09:17 AM
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Actually that 'proof' of yours should go down in history: you assume a and b are distinct numbers, make a false claim about them and use that false claim to prove that a and b a different, which is part of the hypothesis... fantastic


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