Talk:Limit of a function

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To be merged with the main article:

1 Metric spaces
2 Do multiple limits commute?
3 Suggestion
4 non-standard definition
5 sequential continuity
6 Sided limits
7 Interpretation question
8 Definitions of "limit of a function at a point"
9 Display problem
10 Maybe this image will be usefull for EN-WP
11 Derivative page has better information on limits than this page.
12 Limit = -Infinity?

[edit] Metric spaces

The real numbers form a metric space if we use the distance function given by the absolute value: d(x,y) = |x - y|. The same is true for the complex numbers. Furthermore, the Euclidean space Rⁿ forms a metric space with the metric given by the euclidean distance. These three will be our motivating examples for extending the limit definitions given above.

If (x_n) is a sequence in the metric space (M, d), we say that the sequence has limit L iff for every ε>0 there exists a natural number n₀ such that for all n>n₀ we have d(x_n, L) < ε.

If the metric space (M, d) is complete (which is true for the real and complex numbers and Euclidean space, and all other Banach spaces), then one can establish the convergence of a sequence in M by showing that it is a Cauchy sequence. The advantage of this approach is that one need not know the limit in advance in order to do this.

If M is a real or complex normed vector space, then the limit operation is linear, as explained above for the case of sequences of real numbers.

Now suppose f : M -> N is a map between two metric spaces, p is an element of M and L is an element of N. We say that the limit of f(x) as x approaches p is q and write

$\lim_{x \to p}f(x) = q$

if and only if to be merged into the main article:

for every ε > 0 there exists a δ > 0 such that for all x in M with 0 < d(x, p) < δ, we have d(f(x), L) < ε.

This is equivalent to saying

for every convergent sequence (x_n) in M - {p} with limit equal to p, the sequence (f(x_n)) converges with limit L.

The function f is continuous at p if and only if the limit of f(x) as x approaches p exists and is equal to f(p). Equivalently, f transforms every sequence in M which converges towards p into a sequence in N which converges towards f(p).

Again, if N is a normed vector space, then the limit operation is linear in the following sense: if the limit of f(x) as x approaches p is L and the limit of g(x) as x approaches p is P, then the limit of f(x) + g(x) as x approaches p is L + P. If a is a scalar from the base field, then the the limit of af(x) as x approaches p is aL.

If N is R, we can define infinite limits; if M is R, we can define one-sided limits in analogy to the definitions given earlier.

[edit] Do multiple limits commute?

Hi,

One piece of information I was looking for but couldn't find on wikipedia was whether multiple limits commute. That is, given a function $f (x, y)$ , is it necessarily the case that

$\lim_{x \to x_0} \lim_{y \to y_0} f(x,y) = \lim_{y \to y_0} \lim_{x \to x_0} f(x,y)$

? I think this could be added to the article, but I couldn't find it in any of the places I checked online. Thanks! -- Creidieki 05:57, 20 Sep 2004 (UTC)

Limit don't necessary commute. For instance,

$\lim_{y\to0} \frac{x}{x^2+y^2} = \frac1x \mbox{, so } \lim_{x\to0} \lim_{y\to0} \frac{x}{x^2+y^2} \mbox{ diverges;}$

$\lim_{x\to0} \frac{x}{x^2+y^2} = 0 \mbox{ for } y \neq 0 \mbox{, so } \lim_{y\to0} \lim_{x\to0} \frac{x}{x^2+y^2} = 0.$

I believe there is a theorem that the limits do commute in some cases, perhaps somebody else remembers the details. And yes, this would make a good addition to the article. -- Jitse Niesen 15:37, 20 Sep 2004 (UTC)

Here is the only theorem I know of in connection with your question. If the limit

$\lim_{(x,y)\rightarrow (x_0,y_0)} f(x,y)$

exists, then the limits commute. The converse is not true, however. For instance:

$\lim_{x\rightarrow 0}\lim_{y\rightarrow 0}\frac{xy}{x^2+y^2}=\lim_{y\rightarrow 0}\lim_{x\rightarrow 0}\frac{xy}{x^2+y^2}=0$ .

However, the limit $\lim_{(x,y)\rightarrow (0,0)} \frac{xy}{x^2+y^2}$ does not exist.

Moreover, there pretty much can't be any useful non-trivial converse results of the sort Jitse is thinking of since the equality of the commutation of limits is essentially an artifact of the coordinate system. (Exercise for the mathematical reader: But what happens when you apply a homeomorphism to the coordinate system?) Silly rabbit 22:02, 7 June 2006 (UTC)

[edit] Suggestion

I have studied "Calculus AB", a nationwide (US) college standard for Differential Calculus and an intro into Integral calculus. This page, though I could comprehend with some tentativeness, was a struggle to fully understand. Basically what I'm saying is I cannot show this page to someone wanting to learn calculus (with no prior knowledge of limits) and expect them to comprehend it. I would like to work on this page and have added it to my "to do list" but rarely have the time. If someone wanted to add a summary of the concept of limits that would be comprehensible to the average trigonometry/college algebra student (prerequisites for calculus and precalculus), then you've done the job that may take me some period of time. Thanks

In further reveiw, I recommend it be placed near the "examples" section.

I agree; the article is much too formal and textbookish at the moment. Revolver 18:28, 16 September 2005 (UTC)

[edit] non-standard definition

This article appears to give the non-deleted form of the limit as the primary definition, and the deleted version as an "alternative". This is highly non-standard. In almost every real analysis textbook, the limit is taken to mean in the deleted form. Revolver 22:35, 8 September 2005 (UTC)

[edit] sequential continuity

To say that the limit of a function f at p is L is equivalent to saying that for every convergent sequence (xn) in M with limit equal to p, the sequence (f(xn)) converges with limit L.

This is certainly false for general topological spaces, and I am almost certain it is false for metric spaces. Revolver 15:43, 9 September 2005 (UTC)

I correct myself. If a topological space is first countable, then a function is continuous at a point if and only if it is sequentially continuous at that point. Since every metric space is first countable, this statement holds for metric spaces (although it is regarding continuity, not limits...but that is related to my gripe about the recent change in the definition of limit). Since not every topological space is first countable, this statement does not hold for general topological spaces. Incidentally, why is continuity on topological spaces not mentioned? The case of nets does not address this. Nets concern continuous functions INTO a top space, whereas we can also talk about continuous functions FROM a top space. Revolver 16:02, 9 September 2005 (UTC)

I am not aware, off the top of my head, if the equivalence of continuity and sequential continuity characterizes first countable spaces. It's an interesting exercise to consider. Revolver 16:03, 9 September 2005 (UTC)

This is false if we take the deleted form of the definition of limit. For example, consider the characteristic function f of the singleton set {0}. The limit of f at 0, according to the deleted form of the definition, is L = 0, while for the sequence 0,0,0,..., the sequence (f(0),f(0),...) = (1,1,...) converges to 1, not to L = 0. To be consistent, the statement should be modified or the definition should be replaced with the non-deleted form. I prefer the former, because, as Revolver said, almost every textbook takes the deleted form. --Novwik 14:44, 19 February 2006 (UTC)

[edit] Sided limits

There should be a discussion of sided limits of single-variable real-valued functions here. There should also be a discussion of the often published but mistaken lemma that the limit of f(x) as x approaches a is equal to L iff the limit of f(x) as x approaches a from the right is equal to the limit of f(x) as x approaches a from the left (cf Swokowski, Thomas and Finney, et al.). That defect of the intro calculus texts always drives me nuts! The left- or right-sided limit may not exist, but x can still approach a from within the domain. --24.176.68.73 20:59, 25 October 2005 (UTC)

Could you explain more precisely what you mean? I think you may be misinterpreting the definitions. With the usual definitions of limit, and left- and right-hand limits, it is quite true that limit = L iff left-hand limit = right-hand limit = L. If the right-hand limit does not exist, e.g. then there is an epsilon > 0 such that no matter what delta > 0 you choose, [a, a + delta) will have points whose image under the function maps outside the epsilon-band. Then, this epsilon is such that no matter what delta > 0 you choose, you will have points in (a - delta, a + delta) mapping outside the epsilon-band, just by taking the points guaranteed by the above. So, the limit will not exist.

[edit] Interpretation question

In the article it is stated:

* q × ∞ = ∞ if q > 0
* q × ∞ = −∞ if q < 0

Provided that multiplying a finite value by an infinite value is formally valid, I read these statements as:

* Unsigned infinitive multiplied for positive number gives unsigned infinitive.
* Unsigned infinitive multiplied for negative number gives negative infinitive.

Shouldn't it be:

* Unsigned infinitive multiplied for positive number gives positive infinitive.
* Unsigned infinitive multiplied for negative number gives negative infinitive.

Or:

* Unsigned infinitive multiplied for positive number gives unsigned plus/minus infinitive.
* Unsigned infinitive multiplied for negative number gives unsigned minus/plus infinitive.

Or even:

* Unsigned infinitive multiplied for a number gives unsigned plus/minus infinitive.

Am I perhaps misunderstanding this all? [No, I do not have specific knowledge in limits, I'm reasoning with logic, but please, do not trash this comment. After all, if I browse an encyclopedia for 'Limits', it's very possible I do not know anything about Limits; the source of my misunderstanding could be poor explaination of what 'q' is, if we are talking about numbers or results of limits or if ∞ means +∞ or unsigned ∞, e.g.] Thanks in advance.

[edit] Definitions of "limit of a function at a point"

At the beginning of the paragraph, two definitions are given. The second states "Sometimes, the limit is also defined considering for x values different from p.". But it seems to me that the limit is _always_ defined considering for x values different from p: this is implied by "0 < |x-p|" → |x-p| ≠ 0 → x ≠ p. The definition is followed by a formula which seems exactly the same as the one given above. Could someone look into this and give me an opinion?
Stefano85 00:40, 7 January 2006 (UTC)

So what is $\overline{\mathbb{D}}$ suposed to be? Is it a) a closed disk containing p; b) the closure of the decimals? Enquiring minds wish to know. Anyway its non standard notation which should be explained. --Salix alba (talk) 00:22, 18 February 2006 (UTC)

[edit] Display problem

Most of the examples section in this article is not displaying correctly in either Internet explorer or Firefox at present. Elroch 02:23, 16 February 2006 (UTC)

I came to the conclusion the formatting problem was due to extraneous semi-colons, which I removed. The section displays fine now in IE and Firefox on my PC. I hope it's ok on other platforms.Elroch 00:30, 17 February 2006 (UTC)

[edit] Maybe this image will be usefull for EN-WP

[edit] Derivative page has better information on limits than this page.

The Differentiation and differentiability section on the derivative page has a lot of information about limits that this page doesn't, such as the applications and continuity. Someone that actually understands what's going on should probably add these things, as they're fairly important.

[edit] Limit = -Infinity?

I disagree with the one-sided limit example for 1/x which states that the limit equals negative infinity. My calculus book states that such limits that approach arbitrarily large values do not exist, although notation of the type f(x)->Infinity may be used to describe how the limit fails (p. 62-63 of Calculus:One and Several Variables, 8th ed. by Salas, Hille, and Etgen). I am going to "comment out" the part that seems to be incorrect.--GregRM 03:35, 10 December 2006 (UTC)

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