Talk:Limit of a function

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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. — Preceding unsigned comment added by Wshun (talk • contribs) 17:39, 1 August 2003

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,

$\lim _{y\to 0}{\frac {x}{x^{2}+y^{2}}}={\frac {1}{x}}{\mbox{, so }}\lim _{x\to 0}\lim _{y\to 0}{\frac {x}{x^{2}+y^{2}}}{\mbox{ diverges;}}$

$\lim _{x\to 0}{\frac {x}{x^{2}+y^{2}}}=0{\mbox{ for }}y\neq 0{\mbox{, so }}\lim _{y\to 0}\lim _{x\to 0}{\frac {x}{x^{2}+y^{2}}}=0.$

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)

However, the many integral convergence theorems in real analysis (like Lebesgue's Dominated Convergence Theorem) are a kind of limit commutation (if you consider an integral as a limit). — Preceding unsigned comment added by 99.102.100.76 (talk) 11:07, 15 November 2013 (UTC)

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)

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)

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)

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.

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.

Comment: The concept of infinite limit requires a new definition, which can be made in two ways. The first (and most usual) definition posits distinct positive and negative infinite limits, depending on whether f(x) increases in magnitude without bound, while staying positive (+∞) or negative (−∞): this can be phrased in terms of the extended real line. The second way defines a single, unsigned infinite limit ∞, meaning only that the magnitude of f(x) increases, whether positive or negative: this is a version of the projective real line.

The second framework is usual in algebraic geometry but uncommon in real analysis and calculus, so the first is often assumed and the notation simplified by writing just ∞ instead of +∞. I think this is the source of your confusion: just replace ∞ by +∞ in the first pair of equations above. For the second, unsigned framework, the correct equation would be: q × ∞ = ∞ for q any nonzero real number.

A signed infinity could turn into an unsigned one in some special circumstances: for example, if, instead of q, we multiplied by a function q(x) which has lim_x→a⁻ q(x) > 0 and lim_x→a⁺ q(x) < 0 , and another function with lim_x→a f(x) = +∞; then the limit lim_x→a q(x) f(x) = ∞ (unsigned), which we could symbolically write as 0^± × +∞ = ∞. However, this would be a non-standard, confusing symbolism. A variety of similar phenomena are possible, in which signs cancel or confound each other, and it is better just to write the limit statements explicitly. — Preceding unsigned comment added by 99.102.100.76 (talk) 11:55, 15 November 2013 (UTC)

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)

Actually, "sometimes" is correct here because "0 < |x-p|" is not generally accepted as a part of the definition of limit of a function. For instance, in Encyclopedia Britannica only "|x-p| < δ" is used. The "0 < |x-p|" is odd in some contexts as it leads to a definition of continuity that is not 100% congruent with a generally accepted definition based on the concept of topological space. I would speculate that "0 < |x-p|" was injected by some teachers (which would help them to avoid talking about the domain of a function) who were not aware of the topological definition of a continuous function. — Preceding unsigned comment added by 76.171.57.52 (talk) 23:16, 26 May 2015 (UTC)

The topological definition of a continuous function is that the preimage of an open set is open. This is fairly neutral to how one chooses to define a limit. What's actually improved with the deleted definition of a limit is the definition of continuity at a point. There is no corresponding notion for the non-deleted definition of continuity, that you advocate. Most sources of high mathematical provenance bear this out. See, for example, the references I have added to texts by Tom Apostol, Robert G. Bartle, Richard Courant, G.H. Hardy, Walter Rudin. Other references include Calculus by Jerrold Marsden and Alan Weinstein, Michael Spivak Calculus, and basically any undergraduate calculus textbook (although I have the feeling that you would dismiss those as being written by "teachers"), Lars Ahlfors "Complex analysis", Harold Loomis and Shlomo Sternberg "Advanced calculus". In fact, the non-deleted version, while it may be preferred by some (very few) authors, it creates evident convolution in the writing. For example, Jean Dieudonne's "Foundations of modern analysis" uses the non-deleted limits, and he there handles special cases depending on whether the limit point is an element of the set or not. It's not a recipe for clear writing. (And this is the only text of a high mathematical quality that I have been able to find which adopts a non-deleted perspective.) Sławomir Biały (talk) 01:28, 27 May 2015 (UTC)

If I'm not wrong the pioneers of the concept of derivatives got some criticism for dividing with a quantity they later on allowed to be 0 (zero). Division by zero that is. Their defence was that it really wasn't zero. Only just about. This, at first, not so very well defined terrain got, by the non-deleted version, clearly mapped out. 83.223.9.100 (talk) 08:04, 19 November 2021 (UTC)

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)

Maybe this image will be usefull for EN-WP

it's useful but it's need better graphics , i am going to make a better graphics one and post it in the article باسم أس 2 (talk) —Preceding undated comment was added at 23:27, 16 September 2008 (UTC).

The graph should be even easier to read if the f(x) is a monotonically increasing function. Such that p-δ implies L-ε and p+δ implies L+ε. Which is the opposite in the above image. Otherwise this image is excellent. --mgarde (talk) 12:45, 27 September 2011 (UTC)

Limit of a function

This i my attempt at an improvement of the above image. I tried to make a graph that best represents the definition. Comments are welcome. --mgarde (talk) 22:37, 27 September 2011 (UTC)

Wow, this image is amazing. It really shows you what the epsilons and deltas really are in a limit. I suggest putting it up pronto, even while you wait on a better version. Rlinfinity (talk) 14:49, 27 September 2011 (UTC)

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.

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)

The limit is usually considered as unexistant or divergent. Both definitions are acceptable, but since this is a page supposed to be useful for learners, I think it'd be better if listed as divergent (or at least a fair explanation for it being "undefined", because it somewhat beats the concept of "limit".) Linkman 145 22:06, 28 December 2006 (UTC)

Defining the extended real numbers as a topological space with neighborhoods at infinity and -infinity makes it perfectly acceptable to say that the limit of 1/x (as x approaches 0 from the left) is -infinity. Of course, the limit does not exist if 1/x is interpreted to be a real function defined in a metric space. However, using the extended real limit is sufficiently rich so that it's clear that an infinite limit implies an unboundedness. A discussion of the agreement of limit superior and limit inferior might be good here too. --TedPavlic 15:55, 14 March 2007 (UTC)

Check

I just removed the word "wrong!" from the Limit of a function of more than one variable section. Perhaps a knowledgable editor would like to check that section. --zenohockey 05:17, 17 February 2007 (UTC)

Looks right to me now. I think it should be generalized further to more dimensions, so L becomes a vector. Also, what about this definition for $\lim _{x\to a}f(x)=L$ where x, a, and L are vectors?

For every open set N containing L, there exists an open set U, a subset of A, such that (use any of the following in the definition, as they are equivalent):

1. all of U (except maybe a) is mapped into N by f
2. if x is in U, then f(x) is in N
3. f(U\{a}) is a subset of N

Pomte 03:22, 18 February 2007 (UTC)

Hausdorff Space in Definition of Topological Limit

Currently the definition of a limit in a topological space requires that the codomain of the function be a Hausdorff space. This is too strong. Limits are defined in exactly the same way for non-Hausdorff spaces; it's just that Hausdorff spaces happen to have unique limits. So, I suggest that the Hausdorff requirement be removed. --TedPavlic 15:57, 14 March 2007 (UTC)

I had the same thought a while back. But, then we would also have to change "the limit" to "a limit", etc. I think it would be too easy for a casual reader to get the wrong idea. Silly rabbit 17:39, 16 April 2007 (UTC)

Also in order to have a unique limit one does not have to require Hausdorff. US spaces are topological spaces in which every converging sequence has a unique limit. Every Hausdorff is indeed US, but the other way does not hold.109.66.119.138 (talk) 11:22, 6 March 2017 (UTC)

What is a US space? At least "Hausdorff space" refers to something most mathematicians will recognize, even if it is not the most general formulation possible in which there are unique limits. Sławomir Biały (talk) 14:41, 6 March 2017 (UTC)

Limit Point in Definition of Topological Limit

It is not clear to me why a the definition of a limit in a topological space forces the domain point to be a limit point. The neighborhood definition seems sufficiently rich. Additionally, the definition should be weakened so that the function can have a domain that is any subset of the topological space. The function need not be defined for the whole space (minus the point). Any subset will do. However, in that case, the neighborhood must not be disjoint from the domain. --TedPavlic 16:02, 14 March 2007 (UTC)

Correction; the limit point requirement is fine. Otherwise it's possible for isolated points to mess things up. However, I still think the definition can be weakened for all subsets. --TedPavlic 16:08, 14 March 2007 (UTC)

Two things: first, subsets of topological spaces are themselves topological spaces with respect to the induced topology. So my original definition is not weaker than yours, and has been reinstated. (It is simpler.) I mention later how to handle domains strictly included in X. Second, the limit point requirement is necessary, but for the domain of the function (which you had called E), not just for X (of course, this will follow). In fact you seem to use this later with the E ∩ U - {p} ≠ ∅ requirement. Indeed, it isn't hard to cook up an example where limits are non-unique if you relax the limit point condition. Silly rabbit 19:05, 16 April 2007 (UTC)

What's so special about the E ∩ U - {p} ≠ ∅ requirement? The image of the empty set is empty, so the implication is vacuous, and EVERY point is a limit of the function at that point. We aren't afraid of non-unique limits (non-Hausdorff target spaces), what's so intimidating about every point being a limit? 67.181.111.220 (talk) 08:31, 24 September 2009 (UTC)

Moreover, there's good reason TO include isolated points, namely if they are included, the following is true, (if I'm not mistaken): "A function f:X -> Y, top spaces X, Y is continuous at a in X iff f(a) is a limit of f at a." Of course, this nice equiv. condition of continuity cannot be true w/o including isolated points. 67.181.111.220 (talk) 16:42, 24 September 2009 (UTC)

What's so intimidating about multiple limits is that $\lim$ is no longer a function of $f$ and $a$, but a relation between $f$, $a$ and $L$. This means you cannot successfully use the symbol $\lim _{x\to a}f(x)$ in expressions in place of a point in the target space.

You would have to use say $lims_{x\to a}f(x)$ as a set variable and the functions and operations defined in the object space would be unavailable for this symbol because they're defined on the points in the object space, not subsets of the points.

In any case I believe the definition as stated is that generally accepted, so is the one which should appear in Wikipaedia.

Martin Rattigan (talk) 15:57, 30 April 2015 (UTC)

And talk pages exist to discuss improvements on the article, not as fora for discussing the subject. So... Magidin (talk) 20:01, 2 May 2015 (UTC)

Introduction

I altered the intro a bit. The current definition is not quite correct, since it does not capture the "all" part of "for all |x-p|<delta". E.g. the limit of $sin(1/x)$ near 0 would be every number between -1 and 1. The old text is below. Triathematician (talk) 17:34, 20 December 2007 (UTC)

In mathematics, the limit of a function is a fundamental concept in analysis. Informally, a function f(x) has a limit L at a point p if the value of f(x) can be made as close to L as desired, by making x close enough to p. Formal definitions, first devised in the early 19th century, are given below.

In consideration of recent changes, I suggest the following as a new version of the introduction. Triathematician (talk) 18:41, 4 January 2008 (UTC)

In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Informally, a function assigns an output f(x) to every input x. The function has a limit L at an input p if f(x) is "close" to L whenever x is "close" to p. More specifically, when f is applied to each input sufficiently close to p, the result is an output value that is arbitrarily close to L. If the inputs "close" to p are taken to values that are very different, the limit does not exist. Formal definitions, first devised in the early 19th century, are given below.

• Fine with me. Silly rabbit (talk) 18:57, 4 January 2008 (UTC)

Recommend merging with http://en.wiki.x.io/wiki/Limit_(mathematics)

—Preceding unsigned comment added by 67.85.93.45 (talk • contribs)

There are different types of limits; this is just one of them. –Pomte 08:15, 17 March 2008 (UTC)

GA Failed

There are far too few sources for this to meet GA, so I've quick failed it. Ten Pound Hammer and his otters • ^{(Broken clamshells•Otter chirps)} 01:14, 28 May 2008 (UTC)

i saw that there are too resources also when I looked up this page yesterday to use it with a student of mine.. Can I help put some in?? I have several calc textbooks.. (I'm a tutor)aharon42 (talk) 13:06, 6 September 2008 (UTC)

A few ideas for making the concept of a limit even clearer

I know from much teaching experience that the precise concept of the limit of a function is hard to get across to students, especially because the definition involves four separate clauses, all of which must be borne in mind for the concept to be fully understood:

1. For every epsilon > 0,

2. there exists a delta > 0 such that

3. if 0 < |x - c| < delta,

4. then |f(x) - L| < epsilon.

For this reason, I think it would definitely enhance the article if several simple examples were given, with proofs. For instance, the limit of f(x) = 5x as x → 3, and the limit of f(x) = 1/x as x → oo.

I think it would also be helpful to include several examples where the function's value at x = c is different from the limit of f(x) as x → c, and (as mentioned above) several examples where a limit does not exist.

Also, I think a diagram (such as suggested above) with careful annotation would be a great help to beginners.

I think the best way to communicate the concept of the limit of a function "being infinity" is to state the the notation (which of course resembles ordinary limit notation but with the number L replaced by oo) is to describe the notation as shorthand for saying that for any M > 0 there is an delta > 0 such that if 0 < |x-c| < delta, then f(x) > M (and similarly for -oo, as well as altering this as needed for limits where x →oo or x → -oo). This has the advantage of not requiring a definition of oo per se.

Finally, as to the question of whether an infinite limit "exists", I'd recommend that the article take no strong position either way, since it is really a matter of opinion.Daqu (talk) 18:40, 22 June 2008 (UTC)

As someone who is currently attending a mathematical analysis course, I could not agree more. This subject is quite difficult to grasp and few very basic step-by-step examples would highly enhance the articles readability. It seems to me, that you would be an excellent candidate for writing such examples, if you wish to. You could add them here, such that they later could be added to the article, once approved. --mgarde (talk) 12:51, 27 September 2011 (UTC)

Merge?

This article does a good job of stating the definition of a limit. Why is (ε,_δ)-definition_of_limit necessary? Shouldn't it be merged here? Thenub314 (talk) 12:01, 10 October 2008 (UTC)

Yes, that seems like a good idea. siℓℓy rabbit (talk) 12:09, 10 October 2008 (UTC)

I think there was a reason for the split, and it will require careful study, as (ε, δ) now includes both limits of a function and of a sequence. However, I didn't really see a good reason for the split at the time, if I recall correctly. I certainly have no objection to a formal merge request. — Arthur Rubin (talk) 14:49, 10 October 2008 (UTC)

True, we may have to be careful to make sure pages link either to here or limit of a sequence but where we have pages on both of these topics, the page (ε,_δ)-definition_of_limit doesn't seem necessary. Thenub314 (talk) 20:45, 10 October 2008 (UTC)

I vote yes to the merge. This is nothing more than a formal definition of the limit. Besides deserving to be in the "limit" article, it is a necessity for a clear description of the limit. --Hamsterlopithecus (talk) 01:35, 19 October 2008 (UTC)

Another question is what to do with limit (mathematics). There doesn't seem to be much there. siℓℓy rabbit (talk) 03:22, 19 October 2008 (UTC)

The article on limit of function is rather long and does not get to the epsilon-delta definition immediately. The epsilon-delta definition is an identifiable concept immediately recognizable in a community wider than that of mathematicians. As such it should have a separate entry. Furthermore, the epsilon-delta definition is more general than just the framework of functions, as there is one for limits of sequences as well. I think those are three excellent reasons to keep the article separate. Katzmik (talk) 12:26, 22 October 2008 (UTC)

I don't have strong feelings one way or the other, but I do feel that our articles have engaged in much duplication of what can be handled entirely in one place. We have limit of a function, limit (mathematics) (which should probably just be a disambiguation page), and (ε, δ)-definition of limit. Undoubtedly there are other articles that more clearly deserve their own page, such as limit of a sequence. siℓℓy rabbit (talk) 19:16, 22 October 2008 (UTC)

I do not know what the best thing to do with limit (mathematics) would be. I think it should be reworked so that it points to main articles about different notions of limits in mathematics, or perhaps it should be a disambiguation page. I disagree that having an article that immediately jumps into a definition is a reason to have a separate article. Most of the time, it seems to me, the difficult part of writing a good article is including more then just the strict mathematical statements. If the ε-N definition for limit of a sequence were not already in limit of a sequence it would be a good reason not to merge. I also feel for a sequence the title of the article is close to a misnomer. Overall I think limit of a function and limit of a sequence do a better job than (ε,_δ)-definition_of_limit. Thenub314 (talk) 20:03, 22 October 2008 (UTC)

Of course you are right, the articles you mentioned start with an intuitive explanation of the concept, which is always better than a dry definition. However, the (epsilon, delta) article serves a different purpose. Namely, somebody looking for a precise statement of the artual definitions, rather than discussions thereof, will find them conveniently summarized here. As far as duplication is concerned, I think it is an inherent feature of any useful database. In fact, I don't think we have enough of it. Various editors keep complaining that the math articles are not accessible enough. The situation could be corrected to have more articles of the type "Introduction to...". Fine examples are introductions to relativity theory and introduction to quantum mechanics. However, when someone tries to write such an introduction, there is frequently stiff resistance from people worried about forking. I think we should be more worried about accessibility than about forking. Katzmik (talk) 10:37, 23 October 2008 (UTC)

I agree that duplication is not always a bad thing. I also agree we should be more worried about accessibility then forking. In this particular case, I feel the articles limit of a function and limit of a sequence also provide precise statements of the actual definitions and not simply a discussion thereof. Further in the case of both articles the definitions are offset as their own sections and are clearly identifiable from the contents. So if one wants to skip introductory comments it is easy. I think these two articles provide more detail into the precise definitions. Thenub314 (talk) 14:14, 23 October 2008 (UTC)

Good, I am glad we agree. As far as the (epsilon, delta) page is concerned, it is more narrowly defined than the two limit pages you mentioned. Namely, there are more than one way of defining limits, and epsilon, delta is only one of them. An alternative method is provided by non-standard calculus. The (epsilon, delta) page focuses on the Cauchy-Weierstrass approach. As I mentioned already, the main argument in favor of retaining it is that the term is an identifiable and familiar concept, as distinct from the notions of limits themselves. The fact that the epsilon, delta definition is given in a subsection of a larger article is not a reason not to have a separate page on this widely used concept. Having such a page certainly goes in the direction of greater accessibility the desirability of which we seem to agree on. Katzmik (talk) 14:21, 23 October 2008 (UTC)

I am not not a mathematician but like to read math articles. I feel that large duplications are a problem in Wikipedia and should be avoided. If one duplication is expanded or corrected in one article then it is usually not so in the other. The reader may have to read a long section twice. A reader may miss important information that was in the other article. I very much dislike reading though an article and much later finding out that vital information was hidden in an almost identical section in another article.

Lack of accessibility of math article seem often to be a complaint by those who expect to find an ordinary language explanation and do not want to have to read other linked articles. Which is not possible if not every math article would be extremely long and start by explaining formal systems, then first-order logic, then set theory and so on.

It seems very unlikely that anyone would search for an expression with unusual symbols such as "(ε, δ)-definition of limit". If anyone did, then better solutions would be a redirect to the appropriate section or disambiguation page.Ht686rg90 (talk) 17:05, 23 October 2008 (UTC)

Personally I would prefer to have all the material from these four articles, none of which are extremely long and have many duplications, merged into limit (mathematics).Ht686rg90 (talk) 17:38, 23 October 2008 (UTC)

I am not against one article under limit (mathematics), but I am also not against two separate articles on limits of functions and sequences. The title of limit (mathematics) suggests one should include a bit about every notion of limit in mathematics, of which there are many. (For example should we mention weak limits, categorical limits, etc.) But I think duplication in this setting is not needed, so I think some merge should happen. Thenub314 (talk) 18:41, 23 October 2008 (UTC)

How about "limit (mathematical analysis)"? I think that one article for limits of sequences and functions would be beneficial. Then there would be one place for discussing the many similarities such as history and (ε, δ)-definition.Ht686rg90 (talk) 19:14, 23 October 2008 (UTC)

I suppose I do not have strong feelings either way, I can see advantages to ending up with 2 articles with some disambiguation page and also I can see advantages of just having 1 article. I await with interest to see what other people think. Thenub314 (talk) 20:53, 23 October 2008 (UTC)

The page (ε, δ)-definition of limit summarizes the quantifier definition developed by Weierstrass in a succinct way that transcends the dichotomy into limit of function/limit of sequence. It makes it possible for the reader to compare Weierstrass's approach with other approaches, which reduce its quantifier complexity. I already mentioned that the main argument in favor of the page is that it is a widely recognized term and as such deserves a separate page. It is a well-visited page (there were 98 hits yesterday). Katzmik (talk) 08:57, 24 October 2008 (UTC)

""limit (mathematical analysis)" also transcends the dichotomy into limit of function/limit of sequence. Can also explain other approaches. 98 hits is very little for a foundational concept such as limit. Why not simply have a redirect to the appropriate section? Please see the disadvantages I mentioned above with duplications.Ht686rg90 (talk) 09:38, 24 October 2008 (UTC)

The issue of duplication is a separate one and I have a feeling it will be difficult to reach a consensus on this. I personally feel we should worry less about duplication/forking than about accessibility. Another editor above agreed to this. I think it is helpful to have shorter articles. They tend to be more focused and helpful to the reader. My favorite example of this is the page computational formula for the variance that I created, not that I am an expert in statistics or probability. This formula was hidden inside the long page on variance. Since it is a generally recognized term, I felt it should have a page of its own. Numerous editors have concurred, and the page is now considerably longer than when I first created it. I feel similarly about the epsilon, delta page. Katzmik (talk) 09:43, 24 October 2008 (UTC)

I feel maintaining all of these pages limits accessibility. I agree we should be more concerned with it, but feel the current state of affairs hurts accessibility. Thenub314 (talk) 15:49, 26 October 2008 (UTC)

The page at (ε, δ)-definition of limit deals with issues that are not dealt with at the other limit pages. Namely, it deals with the logical structure of the definitions. Given the difficulty of the concept, it seems reasonable to have a detailed discussion of it. Thus, the current section comparing the two notions of continuity in terms of quantifier order is not found at the other pages, and rightfully so, as they deal with other issues. The (ε, δ)-definition of limit page certainly helps accessibility as far as understanding this issue is concerned. Katzmik (talk) 08:46, 27 October 2008 (UTC)

If we are going to expand to an article mentioning and comparing all instances of a (ε, δ)-definition then it should also include topics such as absolute continuity, Riemann integral etc. How about renaming your page to (ε, δ)-definition in mathematics? Regardlesss, to avoid this issue for now, let us just start by merging the other pages to limit (mathematical analysis. If no objection, I will just do a complete merge. Then we can start removing duplications and fix other issues.Ht686rg90 (talk) 10:35, 27 October 2008 (UTC)

I really don't see a consensus for such a merge at this time. If you feel strongly about it, I suggest you raise the issue at WPMath. Incidentally, since I have not had a chance to respond to your comment regarding "corrections and improvements", I would like to mention that frequently quite the opposite is the case, namely a high quality page is made mediocre by a series of well-intentioned but not always well-informed edits. Katzmik (talk) 10:46, 27 October 2008 (UTC)

All your objection have been regarding merging the (ε, δ)-definition page. Please explain your concrete objections to merging the other pages.Ht686rg90 (talk) 10:49, 27 October 2008 (UTC)

This string is getting to be unmanageable. If you don't mind, I will respond at your talk page. Katzmik (talk) 10:51, 27 October 2008 (UTC)

No reason to limit this discussion to my talk page since it concerns this article. I will reply here. I will unindent my answer.Ht686rg90 (talk) 11:06, 27 October 2008 (UTC)

Continued from above. The only concrete reason you give is that few people may have seen this discussion. The other reasons are regarding merging the (ε, δ)-definition page which is not the issue right now. So again, if you have any concrete issues against merging the other three pages then please list them.Ht686rg90 (talk) 11:06, 27 October 2008 (UTC)

There is a difference between a short summary section with a link to the main page and the longer main page, as compared with two long almost identical duplications in two different articles. The first is fine, improvements will automatically be directed to the main page and the interested reader only have to read one long text. If there are two long duplicated texts, then over time there will corrections and improvements distributed randomly between these, the reader will have to read the text twice, or even worse miss one of the texts and some important info only mentioned in one of them.Ht686rg90 (talk) 14:40, 24 October 2008 (UTC)

concrete reasons

Fine, here are two concrete reasons:

1. since the limit concept is one of the central ones in mathematics, drastic changes in the current configuration should be discussed at WPMath and not performed hurriedly, in accordance with the motto "don't fix it if it ain't broke". There is probably a more eloquent Wiki policy to this effect, but I am not as adept at quoting wiki policy as some.

2. Many editors feel that any identifiable concept should have its own page, even at the expense of a certain amount of duplication. Thus, "limit of a function" and "limit of a sequence" are distinct, familiar concepts. I have not looked at limit (mathematics). If it is really true that there is NO material there that's not in the other pages, it could be removed following consensus at WPM. Katzmik (talk) 11:19, 27 October 2008 (UTC)

P.S. I agree with your comment that the page (ε, δ)-definition of limit could be expanded to include additional material. Actually I just looked at Absolute continuity for the first time. It is a concept less familiar from calculus. At traffic statistics, it is visited far less frequently than the other pages discussed above. Having said this, please feel free to add material to (ε, δ)-definition of limit, in line with the basic thrust of that page, namely emphasis on logical structure of Weierstrass's definition and related definitions. Katzmik (talk) 11:35, 27 October 2008 (UTC)

1. Please see WP:BOLD. But we should certainly reach a consensus by discussion.
2. Please also see Wikipedia:Summary style. Wikipedia should in general follow a pyramid structure of briefer summary sections linking to more in depth main articles. Not long duplications of the same material in different articles.
3. Related to 2, if for example there is much material about the (ε, δ)-definition, which may or may not only be restricted to such definitions only for limits, then it should have an article of its own. Personally I would be very happy if you as an expert would expand it to a general discussion of the (ε, δ)-concept in mathematics, not only restricted to limits but also including topics such as continuities and integrals, and maybe change the name accordingly to "(ε, δ)-definition in mathematics" or something similar.
4. Limits in mathematical analysis is a foundational concept so Wikipedia should have a good coverage on this. The three articles limit (mathematics), limit of a function, and limit of a sequence have many duplications. Limit (mathematics) is not a good name since most of the article is about limits in analysis. Disambiguation is already covered in the limit article. Limit of a function and limit of a sequence have many similarities such as history and (ε, δ)-definition better discussed and compared in one article. Thus it would be better to merge these three articles to "limit (mathematical analysis".

The most typical target audience is a high school/college student which may not be extremely interested in mathematics or see it as more than a side subject. Requiring that such readers should read and compare three different articles with many duplications to grasp the concept is not efficient or good writing.Ht686rg90 (talk) 12:25, 27 October 2008 (UTC)

Here are my comments:

Personally I don't find quotations of wiki policy very helpful. Obviously there are situations where one should be bold. Yet there are others when one should stick to "don't fix it". Someone recently quoted a regulation on avoiding the quantifiers. This may be useful as a general directive, but in the context of lengthy definitions involving multiple quantifiers, this is clearly inappropriate.

1. Please also see Wikipedia:Summary style. Wikipedia should in general follow a pyramid structure of briefer summary sections linking to more in depth main articles. Not long duplications of the same material in different articles.

I agree that long duplications are to be avoided. I think the way of dealing with this problem is defining better the goal of each page, as I have tried to do at epsilon, delta. For instance, the lead paragraph in each of limit of function/limit of sequence could elaborate more on what's particular to that case.

1. Related to 2, if for example there is much material about the (ε, δ)-definition, which may or may not only be restricted to such definitions only for limits, then it should have an article of its own. Personally I would be very happy if you as an expert would expand it to a general discussion of the (ε, δ)-concept in mathematics, not only restricted to limits but also including topics such as continuities and integrals, and maybe change the name accordingly to "(ε, δ)-definition in mathematics" or something similar.

This seems like a noble goal. You will grant me the point that I have done most of the work on the article so far. It would be great if others can chip in.

1. Limits in mathematical analysis is a foundational concept so Wikipedia should have a good coverage on this. The three articles limit (mathematics), Limit of a function, and limit of a sequence have many duplications. Limit (mathematics) is not a good name since most of the article is about limits in analysis. Disambiguation is already covered in the limit article. Limit of a function and limit of a sequence have many similarities such as history and (ε, δ)-definition better discussed and compared in one article. Thus it would be better to merge these three articles to "limit (mathematical analysis".

Again, this may be a good idea, but it could be we are overlooking something. My point is that since there is no pressing need for such a change, it should be discussed first at WPM. Do you have any particular reason to objecting to a discussion at WPM?

The most typical target audience is a high school/college student which may not be extremely interested in mathematics or see it as more than a side subject. Requiring that such readers should read and compare three different articles with many duplications to grasp the concept is not efficient or good writing.Ht686rg90 (talk) 12:25, 27 October 2008 (UTC)

Well, on the contrary, because of the weakness of the typical reader, I think it would be preferable to have several short articles even at the cost of moderate duplication of material, than one long one, where everything is perfectly presented but nobody will get that far. After all, the attention span of the typical reader is not what it used to be. Incidentally, I am amazed at how well Bishop Berkeley mastered the material before embarking on his critique. And he did not have the benefit of a 900-page calculus textbook, either, and had to learn everything from primary materials. Katzmik (talk) 12:55, 27 October 2008 (UTC)

I have nothing against discussing this at WPM but would like to understand your objections first. If I understand it correctly it is that you as a general principle want to have articles for "any identifiable concept" even if these are very short and create considerable overlaps? There is nothing specific regarding this merge you object to? If so should not every article in Wikipedia be broken down to what many consider to be short stubs? Like every time a definition or new word or concept is introduced is given there should be separate Wikipedia article? Ht686rg90 (talk) 14:52, 27 October 2008 (UTC)

Obviously I don't mean that every concept should have a separate page, only terms that are widely recognized as "standard" in some sense. Remember our goal is to help readers understand. If there is a term he is already familiar with, with should exploit his familiarity to teach him more about it. It is very difficult to make general rules about this, rather one should proceed on a case by case basis. In this case, I am yet to be convinced that there is a pressing need for drastic changes. On the other hand, the error at absolute continuity should be corrected as soon as possible. Katzmik (talk) 15:04, 27 October 2008 (UTC)

Cannot this be fixed by having redirects from "limit of a function" and "limits to a sequence "to the appropriate section in "limit (mathematical analysis"? If these sections over time become very long then I agree that there should be separate subarticles. But for now merging these 3 articles will not create a very long article due to the considerable overlap.

As a separate issue, any objection to a least renaming "limit (mathematics)" to "limit (mathematical analysis)" to reflect the main content? Disambiguation is already covered in the limit articleHt686rg90 (talk) 15:15, 27 October 2008 (UTC)

Limit (calculus) would probably be better. Most analysts probably don't think of "limit" as a subject in analysis. Katzmik (talk) 15:20, 27 October 2008 (UTC)

I just noticed it's already a redirect :) Somebody already thought of this, and someone else overruled it, I am sure for very important reasons :) Katzmik (talk) 15:22, 27 October 2008 (UTC)

No reason was given for redirect. For example this discussion of mathematical analysis mentions limits prominently.[1] Either name is probably acceptable. Please reply to my arguments regarding redirects and length.Ht686rg90 (talk) 15:53, 27 October 2008 (UTC)

Whew, I think this must be what they mean when they talk about arguing by exhaustion :) Perhaps we can continue tomorrow... Katzmik (talk) 16:00, 27 October 2008 (UTC)

No problem. Note also that the same source does not discuss limits of functions and sequences in separate articles. This allows easy comparisons and discussions of generalizations.[2]Ht686rg90 (talk) 16:03, 27 October 2008 (UTC)

One last comment before I log off: I have seen springer articles before. My impression is often that they are written by experts, are perfectly correct--and unreadable. I have not seen the ones you are referring to, but I would be very dubious about using Springer as a model for wiki. Katzmik (talk) 16:06, 27 October 2008 (UTC)

Just to make sure, is anyone objecting to renaming "limit (mathematics)" to "limit (mathematical analysis)" to reflect the main content? Disambiguation is already covered in the limit article.Ht686rg90 (talk) 11:53, 28 October 2008 (UTC)

I think it should stay and be made more of an overview article with much of the stuff moved out to the separate articles. Dmcq (talk) 12:30, 28 October 2008 (UTC)

But should there not be a general article for limits in mathematical analysis? None of the current articles describe that. The most general concept of a limit would probably be a limit of a filter. Which does not use the (ε, δ)-definition. Thus the (ε, δ)-definition article is not suitable as on overview article for limits since it is a special case. I think if there is an general article then this can describe how all of these concepts are related.Ht686rg90 (talk) 14:07, 28 October 2008 (UTC)

Regarding limit (mathematics), limit of a function, and limit of a sequence, the many duplications would mean that a merged article "limit (mathematical anaysis)" would not be very long. If over time the length increases then subarticles could be created. But for now an article is much easier and more efficient to read than 3 articles with many duplications.Ht686rg90 (talk) 14:19, 28 October 2008 (UTC)

Articles don't need to be very long if they do a useful well defined job. If all those duplications were removed it would become a much better vehicle for pointing out the general idea of a limit and outlining the various types of uses of the idea. It is easier for people to contribute to a shortish article with a straightforward purpose rather than one than rambles all round different things. The difference between wiki and a paper encyclopaedia is the fast linking and I don't think it should be subverted by turning it into a web version of a paper document. Dmcq (talk) 14:41, 28 October 2008 (UTC)

But limit (mathematics) does not have a straightforward purpose since this includes many very difference concepts such as limits in category theory and limit superior. In effect it is only a disambiguation page to very different concepts that can only ramble briefly all round these very different things. We have an article for limits in category theory. Why not an article for limits in mathematical analysis which do would have a straightforward purpose just like the category theory article?Ht686rg90 (talk) 14:56, 28 October 2008 (UTC)

I'm not saying don't have pages on specifics, quite the opposite. I'm saying a disambiguation page specifically for mathematics that said a bit more than is reasonable on a general disambiguation page is a good idea. Otherwise one as to know most of the answer before one can frame a question. Dmcq (talk) 15:57, 28 October 2008 (UTC)

The limit page is already a disambiguation page. Asking that reader should first go to that page, then to another disambiguation page just for math, then have to read 3 other articles, limit of a function, limit of a sequence, and (ε, δ)-definition of limit, including numerous duplications, and still miss many important issues regarding limits which are not discussed in any of them, is not good writing. Adding yet another page with many more duplications for "limit (mathematical analysis)" would make the situation even worse. There is not more material than can fit into two articles, a disambiguation page limit, and an article for limits in analysis. If the material should start to expand, then subarticles can be created.Ht686rg90 (talk) 17:03, 28 October 2008 (UTC)

You're not honestly saying that you think a naive user looking up some concept of limit in mathematics could go to that limit disambiguation page and get any sort of idea where to go next? Just have a look at it. It's just not possible unless you already know already mostly where yo want to go. And it can't be expanded much. They'd have to trawl through the various articles which are quite specific and dive into their matter. Dmcq (talk) 18:00, 28 October 2008 (UTC)

Hi Dmcq, Just wanted to mention a minor point. I am sure you did not mean it this way, but the use of the term "honestly" could be objected to by some people around wiki. It would be best to avoid any kind of misunderstandings of this sort. Katzmik (talk) 18:03, 28 October 2008 (UTC)

Thanks for the tip, I see your point. I certainly had no intention of implying dishonesty in any way, it was an emphasis expression on the impression when asking to look at the article from a naive users point of view. I'll certainly try to look out for expressions of that sort. Dmcq (talk) 18:23, 28 October 2008 (UTC)

limit (mathematical analysis)

Hi, I am not sure why the discussion is continued here, I thought it was moved to WPM but at any rate I would like to address the specific point of the proposed title mentioned in the abstract. I think there is no reason to restrict the limit notion to mathematical analysis. It is certainly used in other fields no less than mathematical analysis. I am a differential geometer more of a topological than analytic bent, and I have certainly used limits in my publications. There is one area where the notion of the limit of of central importance, and that's calculus. Its importance stems, of course, from the fact that it is a tool in defining the basic notions of derivative and integral (its intrinsic value is in fact limited, but that's a separate point). I would support moving the page back to limit (calculus), but I think there is no more reason to move it to limit (analysis) than to limit (algebra). Katzmik (talk) 17:07, 28 October 2008 (UTC)

You are right that we should discuss this in one place and preferably WPM. I will not make further replies here.Ht686rg90 (talk) 17:21, 28 October 2008 (UTC)

A small suggestion for the article

Hi, I'd like to point out that the limit definition in metric spaces is currently not 100% satisfactory. It would be better to require p to be a cluster point of the domain, not only a limit point. In isolated points, indeed, the definition still makes sense, but you lose the uniqueness of the limit since any L can be limit of f (if p is an isolated point there is no other point x to test against, so every L satisfies the condition to be limit). While this is not a major point, it explains why a few authors (e.g. L. Schwartz or E. De Giorgi) preferred a different definition in which you allow d_M(x,p) to be 0 in the definition. Etabeta78 (talk) 07:45, 24 November 2008 (UTC)

I think that lack of uniqueness of the limit is a fairly significant problem with the definition you are proposing. In addition, if you allow for $d_{M}(x,p)$ to be zero in the definition, don't you get something absurd, like a function is continuous at a point if and only if it has a limit there? siℓℓy rabbit (talk) 12:33, 24 November 2008 (UTC)

Wait. First of all, I found out that the current definition of limit is ok (i.e. my suggestion was wrong). The point is that I was considering a slightly different definition of 'limit point' (from one of my calculus courses). I see (from the wiki page for 'Limit Point' ;-) ) that the common definition coincides with the one of 'cluster point' I had in mind. Hence, the current definition in the main page ensures the uniqueness because isolated points are not limit points. My bad.

On the other hand, no lack of uniqueness or 'absurd' result come from the less common definition (which I just mentioned for documentation, I was not suggesting to add it to the wiki page).

The proposition "a function is continuous at a point if and only if it has a limit there" is indeed a consequence of the alternative definition, but it's not wrong or absurd. It's just unusual. Similarly, it is unusual the fact that, with the alternative definition, a function f from R to R such that $f(x)=0$ for $x\neq 0$ and $f(0)=1$ (i.e. constant everywhere but a point) has no limit in 0, even if it has right and left limit (defined as the limit of the function restricted to the open half-lines $]-\infty ,0[$ and $]0,\infty [$). It's mainly a matter of which properties you like to keep ;-)

No point in adding all these details to the main page, however.

(sorry for the missing signature) Etabeta78 (talk) 15:03, 8 December 2008 (UTC)

error in informal definition

I believe the current version of the informal definition in the lead, edited by Dissident, contains the typical error of assuming that the value at p equals L. Katzmik (talk) 07:55, 10 December 2008 (UTC)

History section

Hi, I recently came across 3 references for the history section of this article.

The first would be: "Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus" by Judith V. Grabiner The American Mathematical Monthly, Vol. 90, No. 3 (Mar., 1983), pp. 185-194 http://www.jstor.org/stable/2975545 which attributes the definition to Cauchy.

The second is: "Bolzano, Cauchy, Epsilon, Delta" by Walter Felscher The American Mathematical Monthly, Vol. 107, No. 9 (Nov., 2000), pp. 844-862 http://www.jstor.org/stable/2695743 which points to the work of Bolzano as the first.

Lastly, and the most interesting, (but maybe shouldn't be included is): "Back to Classics: Teaching limits through infinitesimals" by Todor D. Todorov International Journal of Mathematical Education in Science and Technology 32:1 (Jan.-Feb. 2001), pp1-20. Who attributes the first delta epsilon definition to John Wallis in his book "Arithmetica Infinitorium" in 1655 (before Newton and Leibniz work on calculus!)

I find this amusing but don't know what to make of it. The first two would be good additions since the corroborate basically what we have (except Grabiner would seems to make Cauchy's work sound systematic). And it is a printed in a Journal, so it should be a stronger reference then the websites we have now. Thenub314 (talk) 15:07, 17 December 2008 (UTC)

I would remove Todorov's article. It's only two sentences in an article on mathematical education with no references (except to Wallis' book), and I'd like to see further evidence. Furthermore, I don't think that Grabiner and Felscher contradict each other. Both say that Bolzano only gave the epsilon-delta definition for continuity, and that Cauchy a few years later gave the definition for limit and from there, continuity. Would you agree with this interpretation? If so, we should reformulate the sentence about Bolzano. By the way, Victor Katz' book "A History of Mathematics" agrees with Grabiner and Felscher. -- Jitse Niesen (talk) 18:14, 18 December 2008 (UTC)

I am happy to remove the Todorov article, I was never sure if it should be in there in the first place, but I thought I would "give it a try on for size". I agree the do not contradict each other, and they both agree with Burton and now Katz. I am happy to reformulate the sentence about Balzano. I will make a quick attempt now. Thenub314 (talk) 18:57, 18 December 2008 (UTC)

Merge again...

We never quite seemed to reach a consensus on the discussion whether or not to merge. The merge tag was taken out recently and I was a bit surprised. I am still in favor of a merge and would like to revive the debate. My argument would be that the article on (ε,_δ)-definition_of_limit overlaps this article in the following ways. It to gives a shorted form of the history, it gives an intuitive explanation, one of the precise definitions. This article does a better job with these topics. It does talk the importance of the order of the quantifiers, which is important, and I would like to merge into here. Thenub314 (talk) 14:57, 18 December 2008 (UTC)

• Merge. In addition to the unnecessary overlap, the separate article creates the erroneous impression that the ε/δ definition is some special way of defining a limit, as opposed to the standard way. Joriki (talk) 12:07, 26 July 2009 (UTC)

proof required

Please do provide rule to the formulae in the section Useful identities. —Preceding unsigned comment added by VIV0411 (talk • contribs) 16:18, 21 April 2010 (UTC)

Actually, that would be wrong. It's possible we need a reference, but proofs, in general, are not required. — Arthur Rubin (talk) 17:15, 21 April 2010 (UTC)

uniformly continuous

There is a rule about that limits can be pushed through to the argument of a uniformly continuous function, I think this would usefully go close to where limits of arithmetical expressions and indeterminate forms are done. However the only way I've seen this stated is that a uniformly continuous function of a Cauchy sequence is a Cauchy sequence, anyone seen it stated as a tool for evaluating limits? Dmcq (talk) 10:19, 23 July 2010 (UTC)

Evaluating limits at infinity for rational functions

I don't understand why this section exists. There is nothing exceptional about rational functions evaluated at infinities. For interdeterminate forms, use L'Hopital. Angry bee (talk) 01:24, 8 February 2011 (UTC)

Chain rule

I have undone Arthur Rubin's revert of my edit, because he has not, to my satisfaction, provided adequate reasons for his revert. The reason he does give is: 'not "equivalent"; a combination of the first limit and the first condition reads that f is continuous at d.'

This, however, makes little sense to me. Given the fact that $\lim _{y\to d}f(y)=e$ it follows by definition of continuity that the statement f(d) = e is equivalent to saying that f is continuous at d, so what I said was perfectly true. And considering the important nature of continuity within mathematical analysis, I consider it very important to emphasise this fact.Telanian183 (talk) 23:02, 6 April 2011 (UTC)

It's not equivalent. What can be said is that, given $\lim _{y\to d}f(y)=e$, "$f(d)=e$" is equivalent to the statement that "f is continuous at d". The bald statement seems to mean that "$f(d)=e$" is equivalent to "f is continous at d", which is false. I can't think of a way of phrasing the correct statement so as not to be confusing. — Arthur Rubin (talk) 06:52, 7 April 2011 (UTC)

I still don't understand you. You say that what I said above is not correct, and then you repeat it in almost identical words and claim that that is correct! And the fact that we are allowed to assume that the limit in question holds is clearly established by the 'in addition' in the article, so I'm sorry but I really do not follow your reasoning here.

But no matter. I have formulated another way to articulate the same point, which hopefully does not allow any possible room for ambiguity, and have inserted it into the article. — Preceding unsigned comment added by Telanian183 (talk • contribs) 07:52, 7 April 2011 (UTC)

Merge Proposal

I would like to propse we merge parts of the (ε, δ)-definition of limit page that relate to functions to this page. Besides having a name that is very difficult to search for, as most keyboards lack an ε and δ keys. The page is largely repetitive with information contained elsewhere and the sections after the initial informal statement are terse and difficult to read by non-experts. Overall I think merging the two articles into one would improve the quality of our coverage of limits. Thenub314 (talk) 19:19, 19 May 2011 (UTC)

The problem with the special symbols can be solved by moving this to epsilontics, or epsilon, delta. The title involving epsilon, delta (or epsilontics) is more descriptive of the method than a mere reference to limits. Thus, limits are present both in the epsilontic approach and in the non-standard approach (here limit is the standard part of the evaluation at an appropriate nonstandard point, such as an infinite index). What makes Weierstrass's approach distinctive is the use of the quantifier formulas. The notion of limit itself does not make the approach distinctive. Tkuvho (talk) 20:15, 19 May 2011 (UTC)

I have reverted and added back the article. I think it is useful, and not all of the information in it is easily found elsewhere. If the problem is with the symbols in the name, we could just rename the article. I actually think that the "Epsilon Delta Definition" might be the more common way of referring to the definition in writing anyways. And just deleting the whole article really isn't a very helpful way of dealing with parts of the article which might be difficult to understand, if that is the problem.TheFreeloader (talk) 12:22, 17 November 2011 (UTC)

It makes no sense to have a separate article. The epsilon delta definition is something that is (rightly) covered in the main article. (Otherwise about 80% of this article would need to be removed.) Sławomir Biały (talk) 16:57, 17 November 2011 (UTC)

I agree with the merger. It doesn't make sense that there should be an article about the limit and a separate article about its definition. The title is also problematic. NereusAJ (talk) 08:47, 23 December 2011 (UTC)

Given the comments supporting a merger here, [3] and above there seems to be a consensus to merge the two articles. If there are no further objections I will go ahead and do so. Thenub314 (talk) 20:27, 28 December 2011 (UTC)

Limits can also be visualized without the epsilon-delta definition; it is more of a mathematical relief from hopeless situations in weird functions. In other cases, limit laws and L'Hospital Rule are enough. Its best if epsilon-delta is kept as a different topic. — Preceding unsigned comment added by Dwija Prasad De (talk • contribs) 08:47, 13 January 2013 (UTC)

I strongly oppose any merger of this article. As has already been mentioned, delta epsilon is often not used to visualize a limit, to work with limits or calculus. It is a well defined topic and theory of its own that deserves its own page. Millueradfa (talk) 18:48, 24 January 2013 (UTC)

Examles of elementary functions without a limit

I think this could be illuminating for a novice reader, like sin(1/x) at 0 etc., preferably with a picture. Maybe it should be mentioned in the intro. WillNess (talk) 22:51, 18 October 2011 (UTC)

Good idea. Sławomir Biały (talk) 11:14, 19 October 2011 (UTC)

FYI, the article Classification of discontinuities has some images that can be used. Sławomir Biały (talk) 12:38, 29 October 2011 (UTC)

Thanks, I'm apprehensive though to editing here myself, as I'm not a mathematician, and don't have much knowledge of WP machinations (copying images and text from other WP pages etc) either. It'll probably take a considerable time until I read on all the relevant policies etc. Is it allowed for that function from the 3rd case there, text, image and explanations and all, just be copied here? Is it a good candidate you think? WillNess (talk) 22:10, 30 October 2011 (UTC)

It's ok to copy the images here. I think the image captions should be alright too. Sławomir Biały (talk) 11:14, 5 November 2011 (UTC)

Sequential limits and Limit of a function in terms of a sequence

It seems to me that the definitions given for sequential limits and limits of a function in terms of a sequence are incorrect (or at least imprecise): the sequence should not be allowed to take the value a (that is, the sequence should take values in X-{a}; otherwise, a function with a removable discontinuity at a would not have a limit according to the definition (by taking a sequence that is eventually constant a). My attention has been drawn recently to the fact that in France (among other places), the definition of limit does not exclude taking x=a when a is in the domain, following Bourbaki; in which case the given definition would work. But the definition in the text does exclude x=a (which seems to be called a "punctured limit"). Magidin (talk) 03:05, 20 February 2012 (UTC)

Good catch. At some point, I seem to recall that various other bits of the article suffered from the same problem and were fixed. This one obviously slipped through the cracks or was added later. At any rate, it needs to now be fixed as well. Sławomir Biały (talk) 03:40, 20 February 2012 (UTC)

Okay, I've attempted a fix to both sections by specifying the sequence should not take the value a. Perhaps those who are more familiar with the article can clear it up as needed. Thank you. Magidin (talk) 17:05, 20 February 2012 (UTC)

Inputs of known limit

If I have a function and I already know the limit, what is it called when finding the inputs? 68.104.139.226 (talk) 21:13, 28 March 2014 (UTC)

Huh? What do you mean "the limit"? What do you mean "finding the inputs"?Magidin (talk) 01:08, 29 March 2014 (UTC)

New merge discussion

I would like to suggest the possibility of merging (ε, δ)-definition of limit for a few reasons. First, this article has very little that is not already contained in this article. For example they overlap in History, motivation, definition.

While this article can certainly be seen as more general we spend a lot of time discussing various δ,ε definitions. See for example:

• Limit_of_a_function#Functions_on_the_real_line
• Limit_of_a_function#One-sided_limits
• Limit_of_a_function#Functions_on_metric_spaces
• Limit_of_a_function#Limits_involving_infinity
• Limit_of_a_function#Limit_of_a_function_of_more_than_one_variable
• Limit_of_a_function#Relationship_to_continuity

So there is no hope of relegating this material to its own article. Also, this article arguably does a better job the the (ε, δ)-definition of limit. But I would like to hear other peoples thoughts. Thenub314 (talk) 21:54, 13 May 2014 (UTC)

Support, merge and redirect. - DVdm (talk) 08:00, 14 May 2014 (UTC)

Comment Just for reference: As suggested by the section title "New merge discussion", the option of a merge was discussed several times before without reaching consensus. See #Merge? and #Merge again... and #Merge Proposal above, and Talk:(ε,_δ)-definition of limit#more on merge, and Wikipedia talk:WikiProject Mathematics/Archive 66#Suggested Merger of limit pages. Tobias Bergemann (talk) 07:48, 15 May 2014 (UTC)

Support. Tobias Bergemann (talk) 07:49, 15 May 2014 (UTC)

Keep Separate. The (ε, δ) definition is a difficult concept which merits its own page. I express this especially because this page is very difficult to read for anyone that is familiar with only Euclidean space. Jbeyerl (talk) 02:26, 10 July 2014 (UTC)

Support, this article feels highly redundant, and is in need of a lot of work if it will stand as a good article on its own. I agree too that the main limit article ironically does have a better exposition of the epsilon-delta definition, really everything here should be deleted and replaced with a redirect to Limit_of_a_function#Functions_on_the_real_line rather than any effort into merging content 71.225.210.10 (talk) 21:47, 30 November 2014 (UTC)

Support, this article seems a little empty if not mergered with the other pages. I don't see why one fundemantal idea of mathematics needs to be separated, ex. limits of infinity, limits of 0, on and on, when they should be put into one cohesive article named limits. Yes there should be different sections but not separate articles. Right now it looks like information about limits is scattered everywhere in diffrent wikipedia articles when it should be in one Wikipedia article. Doorknob747 22:43, 16 January 2015 (UTC) — Preceding unsigned comment added by Doorknob747 (talk • contribs)

Support: Why should the definition of a subject ever be in a separate article? Providing a detailed definition is pretty much the reason people reference encyclopedias in the first place. I grant that the limit definition might be difficult to understand at first glance, but we could certainly include both "intuitive definition" and "formal definition" sections at the beginning of the article. Jon VS (talk) 04:33, 3 September 2015 (UTC)

I also support this merge proposal: because the epsilon-delta definition is the standard one, any content about it most naturally lives in an article titled "limit". It also seems that there is a pretty strong consensus here in favor of the merge. Is anyone willing to take a careful look at what content in that article is not already included here? --JBL (talk) 18:19, 18 April 2016 (UTC)

• There was unambiguous consensus for a merge back in 2014, and after that several more editors added their agreement, yet nobody did the merge. It seems to me this is astonishingly common: a whole lot of editors agree that merging should be done, yet for some reason none of them actually does it. I shall redirect the other article to this one, and obviously if either any of the participants in the above discussion or anyone else wishes to then merge material from the history of that article to this one, they are free to do so. I don't see anything that is crying out to be merged, so I shall leave it at that. JBW (talk) 19:40, 23 June 2021 (UTC)

He vs. They

I recently edited the "Motivation" section to make it gender-neutral, by replacing "he" with "they". I was reverted, with a reference to WP:MOS. But the MOS doesn't prohibit singular they, as discussed at Wikipedia:Manual of Style/Register#Gender-neutral language. I would appreciate other editors' opinions on this—or perhaps some compromise can be reached that doesn't require the use of either pronoun. —Granger (talk · contribs) 17:30, 16 October 2014 (UTC)

User Slawekb was surely wrong in his edit summary here, and in my opinion the edit was sound. I'd "!vote" for your version. Singular they is okay with me. - DVdm (talk) 18:13, 16 October 2014 (UTC)

The original revision of the motivation section actually used the feminine pronoun, which I have restored. Absent specific direction on the issue of gender-neutral language from the Manual of Style, it is appropriate to keep the conventions used by the original author, rather than to impose our own stylistic preferences in the matter. The link that Mr. Granger gives, although it does not prohibit the use of the "singular they", also does not condone it. And in the discussion linked there, WP:PRESERVE is brought to bear on the matter. Sławomir Biały (talk) 19:15, 16 October 2014 (UTC)

FWIW, I also agree with this . - DVdm (talk) 20:24, 16 October 2014 (UTC)

Thanks for pointing that out, Sławomir Biały. I still prefer using "they" (or replacing the person with something inanimate to sidestep the problem, as in this 2008 revision), but if you two are happy with "she", then I can live with it too. —Granger (talk · contribs) 23:33, 16 October 2014 (UTC)

Same here. - DVdm (talk) 18:31, 17 October 2014 (UTC)

Inconsistency in the limit definitions

The definition of limit for functions defined on subsets of the real line in the section "Functions of a single variable" is not consistent with the definitions in the sections "Functions on metric spaces" and "Functions on topological spaces". In the last two $\lim$ is defined as a function from ${\mathfrak {D}}\subset \{(f,p):\exists \Omega \subset X(f:\Omega \to Y)\land p\in L(\Omega )\}$ to $Y$ (with the notation of the last definition). In the first (using the same notation as opposed to that written) $\lim$ is defined as a function from a rather awkward to describe proper subset of ${\mathfrak {D}}$.

The same notation is used in all cases, which is bound to cause confusion. The results are different. For example

$\lim _{x\to 1}Arcsin(x)={\frac {\pi }{2}}$

using the second and third definitions, but

$\lim _{x\to 1}Arcsin(x)$

is not defined using the first. Again the statement "If either one-sided limit does not exist at $p$, the limit at $p$ does not exist" is true for the first definition but not for the others.

Could the first definition not be revised to correspond with the others?

Martin Rattigan (talk) 18:24, 30 April 2015 (UTC)

misplaced annotation/referring to the wrong definition

In the section "functions of a single variable" right after the definition, it says that the value of the limit not depend on the value of the function at the limit point p in the domain, nor on this function value being well defined. The value of the limit being independent of the value f(p) is true here of course. The limit definition up to that point however is applicable only to functions defined on all real numbers (explicit requirement). So at that point, "the value of the limit" is has no meaning and makes no sense if f(p) is undefined, f(p) has to be defined for a limit to even exist. Later, when the definition is extended, one can say that the value of the limit does not depend on f(p) being well defined. If noone objects, I will move the part referring to the extended definition past the extended definition. Ninjamin (talk) 09:04, 8 May 2019 (UTC)

That's reasonable, although another possibility would be to just modify the first case to not require that p be in the domain of f. Something like, "Let f be a real-valued function defined on all of R, except possibly at p itself." Just the same wording as the second part. This retains the slightly simpler definition for not having to take the intersection with (a, b), but also keeps the generality of not including p in the domain. –Deacon Vorbis (carbon • videos) 13:34, 8 May 2019 (UTC)

Point of accumulation

If f:X -> R, the question arises, for which k does lim_{x->k}f make sense? The notion of k being a point of accumulation of X is useful here. 2A00:23C7:5AAC:801:6DEA:E478:19CA:9623 (talk) 13:37, 1 August 2021 (UTC)

That is included already in the section More General Subsets, using the terminology "limit point". Magidin (talk) 23:32, 1 August 2021 (UTC)

Please elaborate delta-epsilon def. of Lim->infty = infty

In the section on infinite limits, the statement: "These ideas can be combined in a natural way to produce definitions for different combinations, such as

{\displaystyle \lim _{x\to \infty }f(x)=\infty }"

implies that the reader should be capable of extrapolating the given definition for infinite limits at finite domain values, e.g., x->p, p finite, to this case, but due to the use of the expression |x-p| in the latter, such an extrapolation is not obvious: someone please supplement that portion of the article w/ an explicit statement of the "delta-epsilon" definition of Lim as x-> infinity = infinity. 2601:603:4D00:F650:450F:C8E5:B542:66E1 (talk) 05:44, 30 April 2023 (UTC)

Definition of Epsilon and Delta

I feel like prior to their mention on the definition section, the actual terms epsilon and delta are never defined to begin with, just kind of thrown into a statement 137.248.1.6 (talk) 13:42, 20 August 2023 (UTC)

The first occurrence of "delta" was (in section "History") a link to Epsilon, delta which redirects to a later section of Limit of a function. I have replaced the link by "see ... below"; a forward reference may be ok within a "History" section. The text at the link is not particularly well written, but this is another issue, for now. - Jochen Burghardt (talk) 20:05, 20 August 2023 (UTC)

Examples of functions with points for which the limit does not exist

In @Gustamons image under the section of functions of a single variable, shouldn't the graph of f(x)=sinx/x not be included since it has a limit as x->a for all values of a? MGJohn-117 (talk) 03:47, 8 November 2023 (UTC)

@MGJohn-117 Yes, now looking at it, the limit does exist for f(x) = sinx/x at x = 0. I think the image is still appropriate, just the caption needs to be changed to mention that both sides of the limit approach the same value. Gustamons (talk) 05:34, 8 November 2023 (UTC)

epsilon delta vs neighborhood

should we add a short section describing the limit based on the concept of a neighborhood in contrast to the epsilon delta method? It seems like an important omission. DMH43 (talk) 15:40, 27 December 2023 (UTC)

The article already has a section giving a neighborhood-based definition of limits, Limit of a function#Functions on topological spaces. Dmoews (talk) 21:28, 29 December 2023 (UTC)

Gillman removed

I have removed ([4]) the (recently anonymously added) little section Gillman's definition, for the reason given in the edit summary. Plus, it doesn't provide anything new over what is already mentioned in the section Limit of a function#Functions on topological spaces. - DVdm (talk) DVdm (talk) 17:09, 15 June 2024 (UTC)