AoPSWiki Article of the Day
Limit was the AoPSWiki Article of the Day for January 6th, 2008

Hello. I hastily created this article because I noticed we didn't have one on limits. I plan to add things such as proofs for uniqueness, multiplication, and addition, as well as stuff about continuity, left- and right-hand limits, infinite limits, continuous functions, etc., etc., etc. Please contribute if you'd like.

The article looks great. By the way, when you add something to a "Talk" page, add a "signature with time stamp" so everyone knows who said that. The code is two dashes (-) followed by four tildens (~).

--Xantos C. Guin 21:15, 29 June 2006 (EDT)

Oh, sorry; I thought everybody else could see it. That was me, by the way. Also, sorry about that stupid mixing-up of $\delta$s and $c$s. --~~

Whoops; I thought you had written "two tildes." All of the anonymous comments above were written by me. --Boy Soprano II 17:44, 30 June 2006 (EDT)

The "Existence of Limits" section needs work. (In particular, I think $f(x) = \frac{1}{x}$ is a bad example to start with -- $f(x) = \lfloor x \rfloor$ would be better.) Also, there is no mention in this article of infinite limits or limits to infinity, both of which are important. --JBL 14:41, 7 January 2008 (EST)

Regarding the limit of the function $f(x)=0$ for $x \ne 0$and $f(x)=1$ for $x=0$.

This function doesn't have a limit at x=0. Proof: Pick $\epsilon = 1/2$. There is no $\delta>0$ such that $|x-0|<\delta \implies |f(x)-0| < 1/2$ . For any $\delta$ that you pick I can pick $x=0$ and $|f(0)-0|>1/2$. Jep 00:15, 7 May 2008 (UTC)

Your last two sentences are correct, but they have little to do with the first one. For any $\epsilon, \delta >0$ the statement \[0 < |x-0| < \delta \implies |f(x)-0| < \epsilon\] is true. Hence $\lim_{x\to 0} f(x) = 0$. Do you see the difference between this (correct) criterion and yours?

On a different note, I think that this article should not be so rooted in real analysis. There is no discussion of the general, broad idea of "limit", just one narrow application and some details surrounding it. This is unfortunately, particularly as many of the details and adaptations of the reals obscure the main points about limits. Somebody (I?) should probably re-write most of the article and give it a different structure. —Boy Soprano II 02:53, 7 May 2008 (UTC)

I agree that the concept of limit is sufficiently important that it deserves more discussion.

Regarding the example, note that the definition of limit in the main text is different than the one you used above. Definition from main text:

We say that $\lim_{x\rightarrow c}f(x)=L$ iff

$\forall\epsilon>0\;\;\;\exists\delta>0$ such that


The difference is that in the main text you have \[|x-0|<\delta\implies|f(x)-0|<\epsilon\] (I replaced c and L with 0's) and above you used \[0 < |x-0| < \delta \implies |f(x)-0| < \epsilon \quad .\] The zero on the left hand side of the inequality is crucial, no? --Jep 12:06, 7 May 2008 (UTC)

I think it is assumed that $x-c$ is not equal to 0... --1=2 12:48, 7 May 2008 (UTC)

It is correct that in the standard definition of limit, $|x-c| > 0$, but it isn't assumed. It is explicit. The definition in the main text holds only for continuous functions (i think). --Jep 13:00, 7 May 2008 (UTC)

Because of the incorrect definition, when I read the example I didn't actually understand the closing remark: " Note that if our definition required only that $|x-c|<\delta$, the limit of this function would not exist." I would correct the definition and then change that line to something like: " Note that if our definition required only that $|x-c|<\delta$, (as opposed to $0<|x-c|<\delta$) the limit of this function would not exist." Actually I think that if you're writing for high school students it would be better to point out how important it is that the definition of limit doesn't involve evaluating the function {\em at} the limit point, but only {\em arbitrarily close} to the limit point. That was the point of your nice example. The reason for discussing limits in elementary calculus is to cope with quantities that appear to involve "0/0" in their definition. One way of coping is to invoke limits. Another way is to find equivalent definitions that don't involve limits. Personally if I were expanding this page for high school students I wouldn't worry about presenting the notion of limit in its full generalized glory, unless I could make it so clear that the average high school calculus student would find it illuminating. --Jep 21:21, 7 May 2008 (UTC)

Okay, I rewrote parts of the article, in particular the definition. I did put the limit in some of its generalized glory, if not all of it, but for good measure I also included a narrow definition (which isn't that different; it just has some arbitrary and unnecessary constraints). Anyway, I think it's more of a topology thing than a calculus thing, since it has many applications outside of calculus, and in calculus, they're not necesssary—but let's not start that conversation. —Boy Soprano II 21:29, 7 May 2008 (UTC)

Good job. It is greatly improved, but who is the intended audience? If the intended audience is high school calculus students it seems overly formal. They won't know what a metric space is.--Jep 01:04, 8 May 2008 (UTC)

Thanks. That's a kind of good question, which is kind of scary. I guess my intended audience is somebody who is interested in math and either knows what a metric space is; or is willing to look up a metric space is and won't be intimidated by the definition; or is willing to imagine what a metric space might be as they are reading. I also tried to accomodate somebody who doesn't know what a metric space is and just skims or skips all the parts with notation they don't know. I suppose I didn't cover are the people who see the notation and are scared away from the entire article. Hm. —Boy Soprano II 01:30, 8 May 2008 (UTC)

I'm going to restructure it somewhat to make it more accessible. --JBL 14:33, 8 May 2008 (UTC)
Two additional comments: we could keep making this article more and more abstract (one can define limits in arbitrary topological spaces, not just metric spaces, and also in arbitrary categories (although the latter may not be strictly a generalization of the former), but to do that wouldn't give a readable article at all -- I think we could front-load the case of limits of real (or complex) functions even more than I've just done. Second, we definitely need a section about limits to infinity. --JBL 14:50, 8 May 2008 (UTC)

Is there a way to make test pages here that aren't readily accessible to the public? I'd be inclined to try writing something if there were. I don't know exactly what I have in mind but I have a vague idea. I would start with some sort of table of contents that had hyper links to various pages. The table of contents would include things like the following (each would be a separate page): (1)Motivation. Why bother with this gibberish at all? (2)Limits for beginning calculus students. (3)Limits to infinity for beginning calculus students (4)Calculus without limits.(5)Limits on general metric spaces (6)Limits on general topological spaces. (7)??

There'd be a little discussion in the contents explaining who might be interested in each section.

On 4 I would state the basic idea of how to construct the tangent to a point on a curve without using limits and then I'd provide links to the literature for people who are more interested. On 5 I would use what you added yesterday. On 2 I would work off what you had already done and add a bunch more examples. I would have examples of proofs of the limits of various functions at particular points as well as examples of proofs that particular functions don't have limits. In 1 I would probably begin by mentioning that the notion is generally first introduced in a beginning calculus course in order to make sense of the derivative operation and I'd provide a link to that. I'd then go on and explain that many computer languages will return NaN if you try to divide 0 by 0. I'd then discuss the function f(x)=x/x and explain that this function isn't defined at the point x=0 although it is equal to 1 everywhere else. It's a lot of work though. I'm not sure I'd get far enough to make a significant contribution. Somewhere this would need to be tied in to the notion of continuity wouldn't it?--Jep 17:16, 8 May 2008 (UTC)

Well, one reason I like general limits is because is can adjoin $\infty$ and $-\infty$ to the reals to get a new metric space (e.g., $d(x,y) = | \arctan x - \arctan y|$), and then it's just the same old definition. I think that this is something kind of interesting, how the new metric preserves everything we really care about with the old metric even though we change it a lot, whereas I don't think that going through the mechanics of defining limits at infinity are particularly interesting.
Also, this seems kind of like giving up, but should we maybe split the article into a beginning article and an advanced article? —Boy Soprano II 20:33, 8 May 2008 (UTC)

Rather than split it up completely, I think it makes more sense to do something along the lines I suggested above: separate pages that are all linked to under the main "Limit page". --Jep 18:07, 9 May 2008 (UTC)