I dont understand the concept of limit

[wtf]

@Studiot, I admit I'm baffled that you're digging in your heels and denying a totally standard part of math. You're free to do so of course. I have nothing to add to what I've already written.

 

Interested readers should consult the Wiki links I gave, and rest assured that this material is a standard part of math, taught at both the undergrad and graduate level.

 

I'm afraid Professor Hardy is not reachable by mail these days. He died in 1947.

Edited January 25, 2017 by wtf

[Nedcim]

 

 

But no I do not agree that infinity is a limit.

 

Then your in disagreement with various texts and websites that apply that meaning:

So to "become infinite," or to say that "the limit is infinity," is simply the language we use to describe a property of a variable; namely that because its values become larger than any number we might name, they do not approach a limit.

http://www.themathpage.com/acalc/infinity.htm

 

Perhaps you would like to offer a solution to the following proposed limit (or ask Bruce)

Let x_n = n² and y_n = n

Does the limit exist, and if so what is it?

 

Why set up the limit to have an undefined subtraction of infinite limits? Simply, leave in the original form and factor n(n-1) then by the special properties of infinity, infinity times infinity=infinity.

 

I think you are confusing unbounded and infinite.

 

No.

Infinity: the concept of something that is unlimited, endless, without bound.

https://www.britannica.com/topic/infinity-mathematics

[wtf]

Why set up the limit to have an undefined subtraction of infinite limits? Simply, leave in the original form and factor n(n-1) then by the special properties of infinity, infinity times infinity=infinity.

@Nedcim, Your comment sparked something in my brain and I realized that @Studiot has actually made a very good point, one that requires a response.

 

[math]\mathop {\lim }\limits_{n \to \infty } \left( {{x_n} - {y_n}} \right) = \mathop {\lim }\limits_{n \to \infty } {x_n} - \mathop {\lim }\limits_{n \to \infty } y = \infty - \infty = [/math]

@Studiot, You have made a good point here and I apologize for ignoring it earlier.

 

The way I would interpret your question is as follows:

 

Infinite limits appear to violate the theorem that the limit of a sum (or difference) is the sum (or difference) of the limits.

 

 

And you are absolutely right!

 

What is the resolution? Off the top of my head I think we must insist that to invoke this theorem, the limits in question must be members of the real numbers and not the extended real numbers.

 

That is, since the limits in question are infinite, you can not expect to add or subtract them and get a sensible result.

 

I was very curious to see how this is handled in the literature. I got out my dog-eared copy of Rudin's Principles of Mathematical Analysis. At one point he proves the theorem that you can add and subtract limits, but he only proves it when the limits are real numbers. Later on, he introduces infinite limits in the extended reals, but never tries to add or subtract them! In effect Rudin is careful on this point but never calls it out explicitly.

 

It would be interesting to Google around and see whether there's any explicit discussion. The usual limit theorems don't necessarily apply to infinite limits in the extended reals but I don't remember ever seeing that explicitly stated.

Edited January 26, 2017 by wtf

[studiot]

My thanks to both wtf and Nedcim for holding an adult discussion about this subject. +1

 

NedCim, your mathpage link was vastly superior to your previous one and offers some very well presented examples.

 

 

I note they said exactly what I did

 

That a limit is a number and infinity is not a number.

 

and

 

[math]\mathop {\lim }\limits_{x \to \frac{\pi }{2}} \tan x[/math] does not exist

They do offer something of an explanation but Oh dear.

Firstly they say that when we say some limit = infinity we mean something other than conventional 'equals' and the reader needs to know the new conventions.

Better, I say, to avoid such confusion.

They then compound the confusion by introducing two different 'infinities'. (via Gauss' reference to the potential and actual infinities of Archimedes and Eudoxus)

Whoops, what's that ?

We are only allowed one in the extended real number system.

We do not have any rules for two types of infinity.

They also discuss something 'becoming infinite or infinity'

What's that?

Do the rules suddenly change when this happens and what are the rules to know how, if and when it occurs?

In particular if you are going to do some algebra on my question you need to prove which set of rules the n's involved obey and when since n 'becomes infinity' in the limiting process.

Finally they make the classic mistake of claiming that we never reach a limit.

This cannot be true since the definition of continuity at some point, x_o requires that the value of a function equals its limit at x_o.

So to say we can never reach a limit (or that the limit is never included in the set) is stating that we can never reach, say the value of f(x) = x², a x=2.

 

wtf post#28

It would be interesting to Google around and see whether there's any explicit discussion. The usual limit theorems don't necessarily apply to infinite limits in the extended reals but I don't remember ever seeing that explicitly stated.

 

Sorry I didn't make my last explanation very clear.

I will post some background starting from what my sources say was the first formal definition of a limit (Wallis : Arithmetica Infinitorum 1655) going right up to the material you say is hard to find ( Thurston has some vary good pages similar to your Wikipedia article, but without the gloss over the difficult or missing bits)

[wtf]

@Studiot, Perhaps I'm misunderstanding. Are you making a historical point rather than a mathematical one? Wallis, bless his heart, has been dead for over 300 years. He's hangin' with Hardy now.

 

There is no confusion regarding the practice of using the extended reals in modern math.

Edited January 26, 2017 by wtf

[TakenItSeriously]

I'm no expert, but it seems like just another case of a bad analogy, though its basically the same analigy used when I was a student decades ago.

 

Shouldn't a proper analogy include a squared term in order to be valid?

[studiot]

I'm no expert, but it seems like just another case of a bad analogy, though its basically the same analigy used when I was a student decades ago.

 

Shouldn't a proper analogy include a squared term in order to be valid?

 

Some expansion of this would help understand what you are referring to?

[TakenItSeriously]

I'm no expert, but it seems like just another case of a bad analogy, though its basically the same analigy used when I was a student decades ago.

Shouldn't a proper analogy include a squared term in order to be valid?

Some expansion of this would help understand what you are referring to?

Certainly, I was referring to the example used in the OP using the half distance term recursively.

 

I was thinking that it was more appropriate for an example of something like half life.

 

edit to add: or some inverse square relationship.

Edited January 27, 2017 by TakenItSeriously

[wtf]

I was thinking that it was more appropriate for an example of something like half life.

That's an interesting physical example. I'm sure that the concept of half life is not assumed in physics to go on forever. If you keep halving the quantity of something, at some point you can't divide it any more and the process stops. Whereas in math, you can keep dividing a number in half as much as you like. The sequence [math](\frac{1}{2^n})_{n \in \mathbb N}[/math] contains infinitely many distinct terms in math; but only finitely many in computer math or in any physical experiment that can be done, even in theory.

 

Which, by the way, is why the calculus "explanation" of Zeno's paradox fails. Zeno is giving a thought experiment about the physical world, and not about the real numbers as they are presently understood.

Edited January 28, 2017 by wtf

[Nedcim]

@Nedcim, Your comment sparked something in my brain and I realized that @Studiot has actually made a very good point, one that requires a response.

 

 

What am I missing? Simply evaluate the limit:

[math]\mathop {\lim }\limits_{n \to \infty } \left( {{n^2} - {n}} \right) = \infty[/math]

[wtf]

What am I missing? Simply evaluate the limit:

[math]\mathop {\lim }\limits_{n \to \infty } \left( {{n^2} - {n}} \right) = \infty[/math]

Sure, that makes perfect sense. Studiot's point is that we apply the rule that "the limit of a sum is the sum of the limits" then we could split that into the difference of two infinite limits, and it would then be undefined. So we have two different answers for the same problem.

 

The answer is that the rule that the limit of a sum is the sum of the limits does not apply if one or both of the limits are infinite.

[TakenItSeriously]

That's an interesting physical example. I'm sure that the concept of half life is not assumed in physics to go on forever. If you keep halving the quantity of something, at some point you can't divide it any more and the process stops. Whereas in math, you can keep dividing a number in half as much as you like. The sequence [math](\frac{1}{2^n})_{n \in \mathbb N}[/math] contains infinitely many distinct terms in math; but only finitely many in computer math or in any physical experiment that can be done, even in theory.Which, by the way, is why the calculus "explanation" of Zeno's paradox fails. Zeno is giving a thought experiment about the physical world, and not about the real numbers as they are presently understood.

You could use the inverse square law for anything that radiates as well which extends out to infinity.

 

Maybe they're not the easiest concepts to visualize in terms of using analogies which are supposed to be simple examples to follow, but it would make more sense than nonsensical examples.

 

You could always describe them in a way thats easier to understand as well.

 

For example when explaining the inverse square law for anything that radiates, eg light, gravity, magnetism, capacitance, etc.

 

you could imagine a sphere that had a very large but fixed number of N dots evenly distributed all over its surface.

 

As the sphere expands then the number of dots per square meter will fall in proportion to the inverse square of the sphere's radius

n ∝ (1/r²)

where

n represents the dots/square meter

r is the radius of the sphere

 

If N = ∞ then

n will always be approaching but never reach 0

 

n can be whatever the radiating value is, such as the force of gravity or intensity of light

r would be the distance to the center of whats radiating.

 

Edit to add:

I just realized that my first post in this thread was a case of exponential dislexia when trying to find a valid example that matched the OP example.

 

I was confusing 1/n² with 1/2ⁿ

Edited January 28, 2017 by TakenItSeriously

[studiot]

What am I missing? Simply evaluate the limit:

[math]\mathop {\lim }\limits_{n \to \infty } \left( {{n^2} - {n}} \right) = \infty[/math]

 

Well go ahead and do the evaluation please.

 

As written you run straight into the issue that [math]\infty - \infty [/math] is undefined.

 

If you manipulate the expression algebraically to avoid this, you run into the issue that you have to use different rules of algebra for [math]n = \infty [/math] and for [math]n \ne \infty [/math]

 

This is OK if you can guarantee that n is one or the other, but what if you don't know?

 

Here is a more difficult example, where ill considered manipulation can get you into a great deal of trouble.

 

Consider the sum to infinity of the following two series similar to those before.

 

[math]{S_c} = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} + \frac{1}{{32}}.....[/math]

[math]2{S_c} = 2 + 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} + \frac{1}{{32}}.... = 2 + {S_c}[/math]

Which implies

[math]{S_c} = 2[/math]

since

[math]2{S_c} - {S_c} = 2[/math]

But if you try to substitute infinity for Sc and 2Sc you get

[math]\infty - \infty [/math]

Worse, trying this manipulation with

[math]{S_d} = 1 + 2 + 4 + 8 + 16 + 32.......[/math]

leads to

[math]{S_d} = - 1[/math]

Of course the former series is convergent and the latter divergent.

You cannot treat the results of the limiting process the same on a convergent and divergent series etc.

[wtf]

@Studiot, Please explain to us zeta function regularization. Otherwise retract the nonsense in your last post.

 

You've crossed the line from honest questioning to trolling.

Edited January 28, 2017 by wtf

[Nedcim]

 

 

The answer is that the rule that the limit of a sum is the sum of the limits does not apply if one or both of the limits are infinite.

Exactly. If the rule is applied incorrectly as shown above then it will be undefined.

 

 

 

Well go ahead and do the evaluation please.

 

As written you run straight into the issue that [math]\infty - \infty [/math] is undefined.

 

If you manipulate the expression algebraically to avoid this, you run into the issue that you have to use different rules of algebra for [math]n = \infty [/math] and for [math]n \ne \infty [/math]

 

This is OK if you can guarantee that n is one or the other, but what if you don't know?

 

 

As I said earlier, the spacial properties of infinity applies: Infinity times infinity is infinity.

 

[math]\mathop {\lim }\limits_{n \to \infty } \left( {{n} ({n-1}} \right)) = \infty[/math]

 

Do you not guarantee that as n gets large so does the product of n and (n-1)?

 

 

 

 

 

 

 

[studiot]

 

 

Exactly. If the rule is applied incorrectly as shown above then it will be undefined.

 

As I said earlier, the spacial properties of infinity applies: Infinity times infinity is infinity.

 

[math]\mathop {\lim }\limits_{n \to \infty } \left( {{n} ({n-1}} \right)) = \infty[/math]

 

Do you not guarantee that as n gets large so does the product of n and (n-1)?

 

 

 

 

 

 

 

 

 

I'm sorry, where was factorisation of an infinity defined either in your mathpages link or in the Wikipedia reference from wtf?

 

I note you haven't addressed my post#29 about mathpages

[Nedcim]

 

I'm sorry, where was factorisation of an infinity defined either in your mathpages link or in the Wikipedia reference from wtf?

 

I note you haven't addressed my post#29 about mathpages

Again, it's defined by the properties of infinity. It bypasses the undefined issue. Whatever term increases the fastest for large numbers is factored out and that results in a limit that as n approaches infinity results in infinity times 1 as the other terms approach 0. Another property of note is infinity to the power of infinity equals infinity. That shows unless infinity is redefined with a boundary then there is no point in asking what is beyond infinity.

http://www.vitutor.com/calculus/limits/properties_infinity.html

I note you haven't addressed my post#29 about mathpages

True, infinity is not a number.

[studiot]

Again, it's defined by the properties of infinity. It bypasses the undefined issue. Whatever term increases the fastest for large numbers is factored out and that results in a limit that as n approaches infinity results in infinity times 1 as the other terms approach 0. Another property of note is infinity to the power of infinity equals infinity. That shows unless infinity is redefined with a boundary then there is no point in asking what is beyond infinity.

http://www.vitutor.com/calculus/limits/properties_infinity.html

True, infinity is not a number.

 

Unfortunately I get error 404 when I try your latest link.

 

I assume that link was to somewhere that defines the factorisation of infinity?

 

Here is a good discussion about the properties of infinity, including good background on the issues from my post#29 you keep avoiding.

 

https://www.quora.com/What-is-infinity-raised-to-the-power-infinity

 

I note you are also avoiding the difficult bits of my post#38 as well.

Edited January 29, 2017 by studiot

[Nedcim]

 

 

I assume that link was to somewhere that defines the factorisation of infinity?

 

Factoring is allowed in the simplification of a function to compute the limit and that reduces the limit from an undefined to a product type. The property of infinity gives rules for the various products of infinity. All of that is unnecessary because it is understood that the fasting growing term will define the limits at infinity.

 

 

I note you are also avoiding the difficult bits of my post#38 as well.

 

Can you cite these difficult bits I'm avoiding?