Talk:(ε, δ)-definition of limit

Why have symbolic definitions at all?

They seem to really hurt the readability to me. Thenub314 (talk) 07:58, 10 October 2008 (UTC)

You are probably referring to the intrinsic difficulty of the epsilon-delta definition, which is related to the logical complexity of working with multiple quantifiers. I think we are all in agreement that the definition is difficult. Generation after generation of calculus students can attest to this fact, and there is no argument about it at all. On the other hand, there are should be a page that presents such a definition clearly so it can be compared to others. Katzmik (talk) 08:54, 24 October 2008 (UTC)

The definition not using formal symbolic notation is no less clear than the symbolic form, and it can easily be compared to other definitions. I was refering to the Manual of style for math articles that recommends avoiding these symbols. (As do countless others on the subject of writing mathematics.) I was not refering to the difficulty of the definition. Thenub314 (talk) 20:44, 26 October 2008 (UTC)

That's an interesting issue. I suggest we discuss it in the context of the new section I just added, comparing the definitions of continuity and uniform continuity in terms of quantifier order. My argument would be that the comparison becomes far more transparent in terms of the standard quantifier notation used in any honors calculus (in addition to many, many other places that go beyond "service" calculus). Katzmik (talk) 08:34, 27 October 2008 (UTC)

I dispute the claim that "standard quantifier notation used in any honors calculus" is used to give the definition of these concepts in every honors course. At none of the universities I have worked at present this notation for the definition. Many follow Spivak's calculus which makes a point of avoiding this notation. And why should this article limit itself to honors students. I first started trying to read about this definition in high school (for example.) In my opinion (at the level of calculus) formal quantifier notation is mainly used as a convenient black board short hand, and not used by more careful teachers. Thenub314 (talk) 21:41, 27 October 2008 (UTC)

We have wiki guidelines that recommend avoiding technical notation and I agree with the general guidelines. In certain situations following them too literally can be an impediment rather than an aid to readability. As I tried to point out before, the comparison of continuity and uniform continuity in terms of quantifier order becomes more lucid if one can use compact notation. It seems hard to argue with this point. Katzmik (talk) 08:32, 28 October 2008 (UTC)

I disagree it becomes more lucid, and I can argue the point. You may simply write out the two statements as sentences, which every one can read regardless if they are familiar with the symbolic notation. The clause about x and delta still change position. Why is this less lucid? Thenub314 (talk) 13:53, 28 October 2008 (UTC)

OK, I hear what you are saying. I happen to think otherwise--perhaps we will have to agree to disagree on this point. If you think this is an important issue, we can ask for some input from other editors. The truth is, I am not sure how far your reasoning against formulas and in favor of sentences would take you. What about Selberg's trace formula for instance? What about Einstein's mass-energy formula? Katzmik (talk) 17:19, 28 October 2008 (UTC)

I have no difficulty with formulas. The examples you gave express relationships between quantities, which is not really what we are talking about. The point is that, as anyone who has tried to read Bourbaki can tell you, overuse of quantifier symbols makes an article very difficult to read. There are certainly places you cannot do without them. But I just don't think they have a place in this article. Thenub314 (talk) 08:47, 29 October 2008 (UTC)

Not everybody will be happy with your distinction between logical formulas and those in other areas of mathematics. At any rate, you should be aware of the fact that you need to argue your case. Katzmik (talk) 10:08, 29 October 2008 (UTC)

I've left some remarks on the same topic (in favour of words, not quantifier symbols) at WT:WPM, with some comments by other editors. As for the Bourbaki remark, it strikes me as odd: their books are essentially (completely outside of Set Theory?) free of quantifier notation, giving instead an excellent example how carefully crafted language and concepts help replace technical formulas (see, e.g., the wordings of definitions in General Topology, the book most relevant for this discussion). Stca74 (talk) 06:32, 30 October 2008 (UTC)

It does seem I have been unfair to Bourbaki. At some point during grad school I had to look at a book that used quantifier symbols at every place it could. I was not able to get very far through it. I thought the book was by Bourbaki, but I must have been mistaken. I read your remarks at WT:WPM, and I agree with them. Thenub314 (talk) 07:54, 30 October 2008 (UTC)

In his new book, Kevin Houston discusses the difficulty of quantifier definitions. I have not read the book yet. I still feel that in the case of a definition involving four quantifiers, having a compact formula expressing the definition can be an aid to understanding, just as in another field of mathematics a formula is an aid to understanding. Clearly, when quantifiers are used as shorthand in sentences so as to reduce ink outlay, such a practice should be eliminated at wiki. Other than that, I am puzzled by what seems to be the anti-quantifier sentiment. Katzmik (talk) 11:44, 30 October 2008 (UTC)

Actually, in my experience, any symbolism is hard for beginners, but often helpful for those who have worked with the subject for a while. (Historically, the former has been a reason for some medieval mathematical text book authors to avoid employing "complicated abstract notation", such as the Indian/Arabic numerals, or Diophantos's symbolic way to present equations.) I've even (admitted rarely) had calculus students who've asked me to translate statements into quantifier dito.

I find it strange completely to avoid the most common compact way to describe the limit definition. There may be reasons to avoid it in the "gentler" article Limit of a function, but if so, they motivate retaining this article as a separate "more advanced" encyclopedic article, with the neat, short, and very common quantifier definitions included. JoergenB (talk) 20:05, 11 November 2008 (UTC)

The evolution of mathematic notation has moved to maximize "compactness". The quantifier way of writing is an example of precisely that. But in some instances it can be argued that a text reads better by avoiding them. So that's why it is such a good thing to have both: one that reads well and another for the interested who wants to see what it looks like when compacted to the extreme. — Preceding unsigned comment added by 83.223.9.100 (talk) 13:40, 13 January 2017 (UTC)

Confused

In this section it assumes that because e/3 and d is bigger than |x-5| they're equal, why is that?

$0<|x-5|<\delta \ \Rightarrow \ |(3x-3)-12|<\varepsilon .$

Simplifying, factoring, and dividing 3 on the right hand side of the implication yields

$|x-5|<\varepsilon /3,$

which immediately gives the required result if we choose

$\delta =\varepsilon /3.$

200.89.239.231 (talk) 21:27, 14 July 2016 (UTC)

The definition requires the existence of an delta, and the “choose” shoes that. Razcomdary (talk) 11:49, 16 August 2021 (UTC)

Revision

Hello everyone, I just revamped the history section. I intend to add a lot to the entire page as I think it is pretty fundamental! However, I decided to push only my history edits first so that I could get feedback before continuing to everything (FrozenJelloAndReason) 07:47, 10 December 2016 (UTC)

Please put new talk page messages at the bottom of talk pages and sign your messages with four tildes (~~~~). Thanks.

Your edit looks okay, but I have some doubts about the source for the last statement. The source says "Updates on my research and expository papers...". Can you find a more wp:reliable source for this? - DVdm (talk) 09:24, 10 December 2016 (UTC)

I have provided such a source, but I would like to note that I have cheated to some extent insofar as it is just a book created by the author of the original source and contains a lot of revised content from his blog posts.

FrozenJelloAndReason (talk) 02:24, 12 December 2016 (UTC)

Thanks! - DVdm (talk) 17:11, 12 December 2016 (UTC)

For a source contesting Newton's understanding of the limit, see https://halshs.archives-ouvertes.fr/halshs-00655601/document. Here, Ferraro disputes Pourciau's assertion that Newton was the first to present an epsilon argument and argues that Newton's understanding of limits was intuitive in nature (hence not "close" to the rigorous notion formulated by Weierstrass et al.). Perhaps consider including this dissenting viewpoint? 169.237.30.156 (talk) 00:43, 20 September 2018 (UTC)

Limit point

In this section : "If $D=[a,b]$, then the condition that $c$ is a limit point is automatically met because closed real intervals are perfect sets."

The derived set of $D=[a,b]$ is $[a,b]$ itself, so the statement above is only true if $c\in [a,b]$. But since there is no restriction about $c$ (except that it is a real number), the condition is not automatically met. --Rateeyy (talk) 18:19, 22 September 2019 (UTC)

It is hard for me to imagine anyone not understanding exactly what was meant, but just in case I have adjusted the wording slightly to make it more explicit. --JBL (talk) 18:47, 22 September 2019 (UTC)

Why is "N > 0" required in definition of limit at infinity?

The current page has this definition:

Suppose $f$ is real-valued that is defined on a subset $D$ of the real numbers that contains arbitrarily large values. We say that

$\lim _{x\to \infty }f(x)=L$

if for every $\varepsilon >0$, there is a real number $N>0$ such that for all $x\in D$, if $x>N$ then $|f(x)-L|<\varepsilon$.

Why is the $N>0$ restriction is necessary? Is it not enough to say $N\in \mathbb {R}$? For example, we can show $\lim _{x\to \infty }{\Big [}{\frac {1}{x+3}}{\Big ]}=0$. If you pick $\varepsilon =4$, I can pick $N=-11/4$. It's negative but still satisfies the important limit property that if $x>N$ then $|f(x)-L|<\varepsilon$.

Jameshfisher (talk) 19:44, 16 May 2021 (UTC)

Is it not enough to say $N\in \mathbb {R}$? Yes, it is enough (because if there exists a negative N with that property then also every positive N has the property). --JBL (talk) 20:02, 16 May 2021 (UTC)

Probably the $\varepsilon =4$ you chose is too big. Therefore, we did not have to go through the pole, $x=-3$.--SilverMatsu (talk) 04:53, 18 May 2021 (UTC)

Do we prefer to use cited definitions, or "improve" the definitions in the article e.g. by $N\in \mathbb {R}$? Jameshfisher (talk) 08:45, 17 May 2021 (UTC)

It is strongly preferred to use cited definitions -- this is related to the core policy WP:V. (If both versions appear in the literature, then we might include both, with an explanatory footnote.) --JBL (talk) 11:11, 17 May 2021 (UTC)