Talk:Limit of a function

message: 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)
thread_title:  Suggestion 

message: 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)
thread_title: non-standard definition

message: 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)
replies: text: 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)
replies: text: 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)

text: 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)
thread_title: sequential continuity

message: 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 limx→a− q(x) > 0 and limx→a+ q(x) < 0 , and another function with limx→a f(x) = +∞; then the limit limx→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)
thread_title:  Interpretation question 

message: 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)
replies: text: 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)
thread_title:  Definitions of "limit of a function at a point" 

message: 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)
replies: text: 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)
thread_title:  Display problem 

message: 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)
replies: text: 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)
replies: text:    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)
thread_title:  Maybe this image will be usefull for EN-WP 

message: 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)
replies: text: 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)

text: 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)
thread_title:  Limit = -Infinity? 

message: 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)
replies: text: 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)
replies: text:  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)
replies: text:  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)
thread_title:  Hausdorff Space in Definition of Topological Limit 

message: 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)
replies: text: 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)

text: 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)

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

message: —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)
thread_title:  Recommend merging with http://en.wikipedia.org/wiki/Limit_(mathematics)

message: 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)
replies: text:  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)
thread_title: GA Failed

message: 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)
replies: text: 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)
thread_title: A few ideas for making the concept of a limit even clearer

message: 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)
replies: text: 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)
thread_title:  [[limit (mathematical analysis)]] 

message: 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)
thread_title:  error in informal definition 

message: 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)
replies: text: 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)
thread_title:  History section 

message: 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)
thread_title:  Merge again... 

message: 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)
replies: text: 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)
thread_title:  proof required 

message: 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)
thread_title:  uniformly continuous 

message: 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)
thread_title:  Evaluating limits at infinity for rational functions 

message: 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)
replies: text: 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)

text: 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)
thread_title:  Chain rule 

message: 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)
replies: text: 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)

text: 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)
replies: text: 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)

text: 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)
replies: text: Given the comments supporting a merger here,  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)
replies: text: 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)
thread_title:  Merge Proposal 

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

text: FYI, the article Classification of discontinuities has some images that can be used.  Sławomir Biały (talk) 12:38, 29 October 2011 (UTC)
replies: text: 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)
replies: text: 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)
thread_title:  Examles of elementary functions without a limit 

message: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Sequential limits and Limit of a function in terms of a sequence 

message: 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)
replies: text: Huh? What do you mean "the limit"? What do you mean "finding the inputs"?Magidin (talk) 01:08, 29 March 2014 (UTC)
thread_title:  Inputs of known limit 

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

text: 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)

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

text: 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)

text: 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)

text: 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)

text: 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)

text: 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)
thread_title:  New merge discussion 

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

text:  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)
replies: text:  Same here. - DVdm (talk) 18:31, 17 October 2014 (UTC)
thread_title:  He vs. They 

message: 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)
thread_title:  Inconsistency in the limit definitions 

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

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

message: 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)
thread_title:  Please elaborate delta-epsilon def. of Lim->infty = infty 

message: 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)
replies: text: 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)
thread_title:  Definition of Epsilon and Delta 

message: 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)
replies: text: @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)
thread_title:  Examples of functions with points for which the limit does not exist 

message: 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)
replies: text: 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)
thread_title:  epsilon delta vs neighborhood