I dont understand the concept of limit

[giordano bruno]

Our maths teacher was explaining the other day limits and put an example of a turtle that advances half of the way each time

The turtle runs one meter, then half meter, then a quarter of meter... so the turtle limit is 2 without never ever reaching it

Several costudents were questioning the teacher lesson, most would say they thought the turtle would reach destination

After the class a friend gave a solution to this:

The turtle advances one meter, in one second, half meter, in half second quarter of meter, in a quarter of second... so obviously by second 2 will have reched destination and by second 3 will have traspass it

Seems the same to me with mathematical limits:

You add 1 in one second, 0.5 in half second 0.25 in quarter of second

So by two seconds adding you will have added up to 2 and by 3 seconds you will have added up to 3

This limits thing is like i set my limit on two, all right but i set as the turtel my own limits its not like i say now every one of your unit of distance, each meter, i divide it by a number that tends to zero so i cursed you into a restricted space of 1 m since you can not go beyond it

I dont know the english name for this kind of thinking in spanish we call it pajas mentales

[wtf]

The idea of a limit is that something (a sequence or a function) gets arbitrarily close to some value.

 

Nothing is said about "reaching" the limit.

 

So for example the sequence [math]\frac{1}{2}, ~~\frac{1}{4}, ~~\frac{1}{8},~\dots ~~[/math] gets arbitrarily close to [math]0[/math]. That means it gets as close as you want to zero.

 

No term of the sequence is ever zero, and we do NOT talk about it "reaching" zero because that makes no mathematical sense in this context.

 

It's true that we might INFORMALLY think that, but that is not the same as formally defining a limit. And until you understand what a limit is, it's counterproductive to think about it as "reaching" zero, because that makes it harder to understand the actual meaning of limit.

Edited January 21, 2017 by wtf

[MigL]

Your friend's solution doesn't describe a limit, but is more closely related to Zeno's paradox.

If the turtle advanced half the distance for each subsequent second, it will need an infinite amount of seconds to reach twice the distance.

Since this is an impossibility, we say that the limiting value is 2 as the time approaches infinity.

This is a limit.

[studiot]

Hello Bruno and welcome to ScienceForums.

 

I'm sorry the site was down yesterday afternoon so my earlier answer was lost.

 

It would be useful learn what you already know, since limits are very very important and many struggle at the beginning.

Limits are usually a student's first encounter with infinity.

Is this High School level?

Do you know what a 'set' is in mathematics?

 

 

As already mentioned, your teacher's example was one of Zeno's paradoxes.

Zeno was an Ancient Greek and they did not understand limits in their mathematics.

 

https://www.google.co.uk/?gws_rd=ssl#q=zeno%27s+paradoxes

 

To start here are some important facts about limits.

 

Firstly a limit is not a property of a single number. It is a collective property of a whole bunch of numbers.

 

We call a bunch of numbers a set of numbers.

 

So the number 1 or 2 or zero are not, by themselves limits.

 

But when we introduce a whole bunch of numbers for example the reciprocals of the counting numbers and place them in (decreasing) order

 

[math]\left\{ {\frac{1}{1},\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5}.......} \right\}[/math]

We see that the collection gets closer and closer to zero, the further we go along the list.

We say that the limit of this sequence is zero.

But waitup. Consider a different sequence of the same numbers

[math]\left\{ {1,2,3,4,5............} \right\}[/math]

This sequence gets larger and larger the further we move along it.

We say the first sequence is convergent and the second divergent.

Such Divergent sequences do not have a limit.

We say that the limit does not exist.

So a limit may or may not exist for a given situation, depending upon the circumstances.

An important requirement is that the limit be the same sort of mathematical object as all the others in the set or sequence.

Infinity is not a number.

Note that the limit number does not have to belong to the set or sequence.

In the first example zero is not in the sequence.

If either of these are the case then we can never reach the limit. That is the limit is not accessible.

But if the limit number is an element of the set or sequence then we can reach it.

It is not true to say that we can never ever reach a limit.

Sometimes we can, sometimes we can't.

Sometimes even stranger things can happen, but these are best shown graphically and I will leave that part to next time if you are still interested.

Also our set need not be about numbers but may be about any suitable mathematical object.

For example as we draw regular polygons of an increasing number of sides, the closer the polygon gets to a circle.

Being geometers, the Ancient Greeks did explore this aspect of limits and came up with a respectable value for [math]\pi [/math].

 

[Sriman Dutta]

 

But waitup. Consider a different sequence of the same numbers

This sequence gets larger and larger the further we move along it.

We say the first sequence is convergent and the second divergent.

Such Divergent sequences do not have a limit.

We say that the limit does not exist.

The limit is infinity.

[studiot]

The limit is infinity.

 

Not at all.

 

The sequence 'increases without limit'

 

Which roughly means that consider any proposed limit, m then take n>m and the series has a term n which is greater than the proposed limit m which is a contradiction.

 

So the sequence has no limit.

 

 

 

The OP has been banned but since the subject is of such fundamental importance we can continue to discuss it if you wish.

[Nedcim]

 

Not at all.

 

The sequence 'increases without limit'

 

Which roughly means that consider any proposed limit, m then take n>m and the series has a term n which is greater than the proposed limit m which is a contradiction.

 

So the sequence has no limit.

 

 

 

It is not wrong to say a that sequence diverges to infinity only quicker to label it as divergent.

A sequence that does not converge. For example, the sequence 1, 2, 3, 4, 5, 6, 7, ... diverges since its limit is infinity (∞).

http://www.mathwords.com/d/divergent_sequence.htm

[studiot]

It is not wrong to say a that sequence diverges to infinity only quicker to label it as divergent.

 

And your point is....?

[Sriman Dutta]

 

Not at all.

 

The sequence 'increases without limit'

 

Which roughly means that consider any proposed limit, m then take n>m and the series has a term n which is greater than the proposed limit m which is a contradiction.

 

So the sequence has no limit.

 

 

 

The OP has been banned but since the subject is of such fundamental importance we can continue to discuss it if you wish.

But, isn't infinity a limit ?

[studiot]

But, isn't infinity a limit ?

 

No, it fails on several criteria.

 

We were discussing numbers and infinity is not a number.

A limit has to be of the same type of mathematical object as that being 'limited'

Note it does not have to be a member of the set of elements but it does have to be of the same type.

I already noted this in post#4.

 

Further the meaning of the word is 'without limit' or 'limitless' or endless.

 

Another way to look at it might be to think of a limit as a stop but there could be more elements (numbers) beyond the limit, if it is a member of the set of elements considered.

 

So [math]\mathop {\lim }\limits_{x \to 0} x:x \in N = 0[/math]

That is if we approach the limit of a sequence of negative integers it is zero, but there are all the positive integers beyond zero.

What do you think is beyond infinity?

[Sriman Dutta]

 

 

No, it fails on several criteria.

 

We were discussing numbers and infinity is not a number.

A limit has to be of the same type of mathematical object as that being 'limited'

Note it does not have to be a member of the set of elements but it does have to be of the same type.

I already noted this in post#4.

 

Further the meaning of the word is 'without limit' or 'limitless' or endless.

 

Another way to look at it might be to think of a limit as a stop but there could be more elements (numbers) beyond the limit, if it is a member of the set of elements considered.

 

So [math]\mathop {\lim }\limits_{x \to 0} x:x \in N = 0[/math]

That is if we approach the limit of a sequence of negative integers it is zero, but there are all the positive integers beyond zero.

What do you think is beyond infinity?

 

Hmm.. Thanks studiot for making it clear.

[studiot]

I've noted a few things about limits they don't tell you at the beginning but I think help understanding.

There's quite a lot more to pull it all together if you want to go on.

When you can see the whole picture and how it ties in with the rest of analysis, it may suddenly make much more sense.

Edited January 23, 2017 by studiot

[wtf]

What do you think is beyond infinity?

What an interesting question.

 

What do you think is beyond infinity?

 

Do you include the transfinite ordinals and cardinals as being beyond infinity? How about the so-called large cardinals studied in set theory? These are cardinals so big they can't be proved to exist in standard set theory.

 

Hope this isn't too much of a thread jack but when someone asks what's beyond infinity ... that's a very thought-provoking and interesting question.

 

Actually you can put the order topology on ordinal numbers and then talk about limits. It's possible to have a limit point of a topological space that can't be reached by any sequence. So this is not completely off topic. There are limits that do go "beyond infinity," or at least far beyond the natural numbers.

 

https://en.wikipedia.org/wiki/Large_cardinal

 

https://en.wikipedia.org/wiki/Order_topology

Edited January 24, 2017 by wtf

[studiot]

What an interesting question.

 

What do you think is beyond infinity?

 

Do you include the transfinite ordinals and cardinals as being beyond infinity? How about the so-called large cardinals studied in set theory? These are cardinals so big they can't be proved to exist in standard set theory.

 

Hope this isn't too much of a thread jack but when someone asks what's beyond infinity ... that's a very thought-provoking and interesting question.

 

Actually you can put the order topology on ordinal numbers and then talk about limits. It's possible to have a limit point of a topological space that can't be reached by any sequence. So this is not completely off topic. There are limits that do go "beyond infinity," or at least far beyond the natural numbers.

 

https://en.wikipedia.org/wiki/Large_cardinal

 

https://en.wikipedia.org/wiki/Order_topology

 

Yes it is an interesting question and not yet fully resolved.

 

You need to study the Continuum Hypothesis from Cantor to Cohen for this.

 

https://www.google.co.uk/?gws_rd=ssl#q=cohen+and+the+continuum+hypothesis

 

Edited January 24, 2017 by studiot

[wtf]

I need to study CH to understand how nets generalize the concept of limits? Studiot you did not read my post. And if you've been Wiki surfing, surely you know that work on CH is far past Cohen these days.

 

I must say I'm a bit annoyed that you quoted my post without actually engaging with any part of it. CH truly has nothing whatsoever to do with what I wrote nor with any aspect of the generalized theory of limits.

Edited January 24, 2017 by wtf

[studiot]

I need to study CH to understand how nets generalize the concept of limits? Studiot you did not read my post. And if you've been Wiki surfing, surely you know that work on CH is far past Cohen these days.

 

I must say I'm a bit annoyed that you quoted my post without actually engaging with any part of it. CH truly has nothing whatsoever to do with what I wrote nor with any aspect of the generalized theory of limits.

 

Far from being annoyed, perhaps you should read the earlier post questions and replies more thoroughly.

 

I was trying to develop a consistent presentation starting from pretty elementary stuff.

 

I had not even reached the stage of epsilon-delta.

 

I was also avoiding the fact you seem to me to have implied that a 'limit' can never be reached

 

 

wtf post#2

Nothing is said about "reaching" the limit.

 

When of course some limits are accessible and some are not.

 

Both of these are in the next stage of the development.

[wtf]

I'm sure we have an interesting conversation here if we can figure out what it is. You know that if you ask "what's beyond infinity" that's the kind of bait I can't resist. What happened to the OP? Their question seemed perfectly reasonable.

Edited January 24, 2017 by wtf

[studiot]

I'm sure we have an interesting conversation here if we can figure out what it is. You know that if you ask "what's beyond infinity" that's the kind of bait I can't resist. What happened to the OP? Their question seemed perfectly reasonable.

 

Posts 472 and 473 here

 

http://www.scienceforums.net/topic/29763-bannedsuspended-users/page-24#entry968109

 

 

I was trying to help other members who are also struggling with elementary limits as many do.

 

Help in such a venture is always appreciated.

[EdEarl]

@giordano

There is a simple way to understand limits, for example a car can only go so fast on a level road, no wind, and other conditions (e.g., we can say car X is limited to 139km/hr on a level road, no wind, 30% humidity, 20C, ...). There may be a legal speed limit on a road, which means you can go up to that speed without risking a ticket. On the other hand, the speed of light is an absolute limit. A mass may get closer and closer (gamma) to the speed of light, c, where c-gamma=epsilon and epsilon>0. Epsilon can be very small (i.e., it can approach 0) but cannot equal 0. Only pure energy can go the speed of light.

Edited January 25, 2017 by EdEarl

[Nedcim]

 

And your point is....?

That despite what you noted earlier a sequence that is divergent can in fact have a limit that is infinity. Are you in disagreement with that point and the cited link?

 

 

What do you think is beyond infinity?

 

Infinity. Infinity is a concept by definition that has no bounds, It's only in special hypotheses where the general rules of infinity are violated there can be any consideration of what is beyond infinity.

[studiot]

That despite what you noted earlier a sequence that is divergent can in fact have a limit that is infinity. Are you in disagreement with that point and the cited link?

 

Infinity. Infinity is a concept by definition that has no bounds, It's only in special hypotheses where the general rules of infinity are violated there can be any consideration of what is beyond infinity.

 

First statement

Yes, of course a sequence can diverge to infinity.

 

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

 

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?

 

 

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

Second statement.

I think you are confusing unbounded and infinite.

They have different meanings.

An infinite sequence can be unbounded

eg 0,1,2,3........

or bounded

eg 0,1,0,1,0,1..................

Edited January 24, 2017 by studiot

[wtf]

@Studiot, You know about the extended reals, right? Those are the reals with symbols [math]\infty[/math] and [math]- \infty[/math] adjoined. Their purpose is just to make it possible to talk about limits at infinity and infinite limits.

 

https://en.wikipedia.org/wiki/Extended_real_number_line

 

In particular, it's perfectly reasonable to write, say,

 

[math]\lim_{x \to 0^+} \frac{1}{x} = \infty[/math]

 

Wikpedia has an explicit discussion of this here. https://en.wikipedia.org/wiki/Limit_of_a_function#Infinite_limits

 

They make the point that the above limit equation should be read "increases without bound" or some such; and that alternately, we can introduce the extended reals so that we can legitimately talk about a limit being infinite.

 

The reason it's convenient to talk about infinite limits is to distinguish the two different meanings of "diverge." The sequence [math]1, 2, 3, \dots[/math] diverges (in the reals) in a very different way that the sequence [math]0, 1, 0, 1, \dots[/math] does. In the former case, we can say that [math]1, 2, 3, \dots[/math] converges in the extended reals to [math]\infty[/math].

 

It's just a semantic point, but it's (to the best of my knowledge) fairly standard. In other words we're not saying anything profound, we're just introducing some notation and terminology for convenience.

 

And also, a common point of confusion, these sysmbols [math]\infty[/math] and [math]- \infty[/math] have absolutely nothing to do with the transfinite ordinals and cardinals of set theory.

Edited January 25, 2017 by wtf

[studiot]

@Studiot, You know about the extended reals, right? Those are the reals with symbols [math]\infty[/math] and [math]- \infty[/math] adjoined. Their purpose is just to make it possible to talk about limits at infinity and infinite limits.

 

https://en.wikipedia.org/wiki/Extended_real_number_line

 

In particular, it's perfectly reasonable to write, say,

 

[math]\lim_{x \to 0^+} \frac{1}{x} = \infty[/math]

 

Wikpedia has an explicit discussion of this here. https://en.wikipedia.org/wiki/Limit_of_a_function#Infinite_limits

 

They make the point that the above limit equation should be read "increases without bound" or some such; and that alternately, we can introduce the extended reals so that we can legitimately talk about a limit being infinite.

 

The reason it's convenient to talk about infinite limits is to distinguish the two different meanings of "diverge." The sequence [math]1, 2, 3, \dots[/math] diverges (in the reals) in a very different way that the sequence [math]0, 1, 0, 1, \dots[/math] does. In the former case, we can say that [math]1, 2, 3, \dots[/math] converges in the extended reals to [math]\infty[/math].

 

It's just a semantic point, but it's (to the best of my knowledge) fairly standard. In other words we're not saying anything profound, we're just introducing some notation and terminology for convenience.

 

And also, a common point of confusion, these sysmbols [math]\infty[/math] and [math]- \infty[/math] have absolutely nothing to do with the transfinite ordinals and cardinals of set theory.

 

 

OK so let us try to solve my question using the extended real number system.

 

Would you agree with me that the symbol [math]\infty [/math] must always stand for the same thing, just as say the symbol 9 must always stand for the same real number?

 

So using this terminology

 

[math]\mathop {\lim }\limits_{n \to \infty } {x_n} = \mathop {\lim }\limits_{n \to \infty } {n^2} = \infty [/math]

and

[math]\mathop {\lim }\limits_{n \to \infty } {y_n} = \mathop {\lim }\limits_{n \to \infty } n = \infty [/math]

Substituting

[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]

What is the infinite symbol take away the infinite symbol ie what is (the same thing) take away (the same thing) ?

And why is it not zero?

Because I think the 'limit' should actually be infinity.

The penalty you pay for using this extended number system is that normal arithmetic no longer works.

You cannot guarantee that sum/product/difference/quotient will yield a sensible result

You cannot guarantee the axiom of associativity.

Worse.

The point/power of convergence is to determine if there are any solutions to differential equations/complex integration/transformations etc and once you let indeterminate elements into your fold you loose this ability.

Yes I said somewhere back that some authors do this (Professor Thurston for instance), but associating limits only with convergence gains you far more than it looses.

Edited January 25, 2017 by studiot

[wtf]

@Studiot, Just to be clear, you personally reject this entirely standard and common piece of math?

 

How do you get measure theory off the ground? What's the measure of the real numbers?

 

https://en.wikipedia.org/wiki/Extended_real_number_line#Measure_and_integration

 

Going further, if the Lebesgue measure of the real line is not [math]\infty[/math], and you still agree that the measure of a line segment of length [math]1[/math] is still [math]1[/math], then you have to abandon countable additivity. And now you just lost measure theory, which means you lost most of functional analysis, quantum physics, and a lot of other good stuff.

 

What say you?

Edited January 25, 2017 by wtf

[studiot]

@Studiot, Just to be clear, you personally reject this entirely standard and common piece of math?

 

How do you get measure theory off the ground? What's the measure of the real numbers?

 

https://en.wikipedia.org/wiki/Extended_real_number_line#Measure_and_integration

 

Going further, if the Lebesgue measure of the real line is not [math]\infty[/math], and you still agree that the measure of a line segment of length [math]1[/math] is still [math]1[/math], then you have to abandon countable additivity. And now you just lost measure theory, which means you lost most of functional analysis, quantum physics, and a lot of other good stuff.

 

What say you?

 

Let me just say I find Wiki disingenuous.

 

 

Wiki

In general, all laws of arithmetic are valid in R' as long as all occurring expressions are defined.

 

All the laws?

 

 

Wiki

Arithmetic operations

All the rest of this paragraph is unreproducible here

 

Too many things are undefined or omitted in the gloss over treatment.

 

 

Wiki

With these definitions R' is not even a semigroup, let alone a group, a ring or a field, like R is one. However, it still has several convenient properties:

 

Without this most of useful mathematics falls apart.

 

Well I'm sorry I can't even quote from your Wiki link here I just had some stupid error message.

I've just goT through changing their stupid image types for ordinary letters.

 

So much for standard.

 

Talking of standards, perhaps you would like to write to Professor G H Hardy, Cambridge University and tell him to amend his book,

 

A Course in Pure Mathematics

 

To conform to you standards.