Talk:Asymptote

how to
ive said it once ill say it again, these math articles should include ways to find whatever they are talking about. for instnace, this ine should show HOW to find a horizontal asymptote or a vertical. honestly, who is coming here to find a definition that uses complex words and concepts noone but mathematicians understand? keep it simple and show HOW.--Jaysscholar 02:31, 20 October 2005 (UTC)

totally

I agree. I was looking for a less technical illustration.

When I look at the definition of a mathematical word, I don't generally want to be blunted right on with the complex ideas. Instead, keep is simple at first, and give us some time to breathe. ---Wanlei

oldephebe - theautumnfirecdproject.com

It takes all kinds to make a world. In answer to Jaysscholar's question, I'm a math teacher who advised his students in Calculus class that an asymptotic function can approach the straight-line limit from two sides. I got out a Webster's dictionary and read it to them. They said that my answer conflicted with the back of the book.

Now i have a (fairly) authoritative source which implies I was correct. If anything, I would like the entry to say that monotonic asymptotic functions are to be distinguished from those that are simply asymptotic, which are not necessarioy monotonic. mmeo

This entry on asymptote is illegible to the common reader. A new entry is needed to supplement the technical definitions.

I agree with the last comment, the entry is mathematically correct but clear as mud! My suggestion is to have an idiot's definition somewhere near the top for instance "an asymptote is the line a function never quite reaches" - M Dearman


 * I agree with the last two comments. I won't go so far as to substitute Dearman's definition for the one in the article (although I like it), but is there any objection to at least deleting the words "arbitrarily closely" from the definition in the article?  It not only makes it hard to understand, it's also incorrect.  (To me, at least -- but I'm just a layperson.  Maybe I'm missing something?)  As far as I can tell, there's nothing "arbitrary" about the distance of the curve from the asymptote at any given location.  The point's location is given by the function.  -- M.C.

'arbitrary' has a slightly different/more precise meaning in mathematics. It doesn't mean 'random', but rather that the statement is true for any value you can choose, i.e, a function is, for some point, as close to the asymptote as you like. For all x not equal to 0, the function has a value x away from the asymptote. — Preceding unsigned comment added by 111.69.224.157 (talk) 07:51, 23 August 2013 (UTC)

I am STRONGLY in favour of having a "How To" on this page!! OI am currently in my last year of highschool and came to this webpage specifically for that. To my dismay there wasn't a "How To" section. PLEASE won't someone add one SOON! Thanks Plenty.. TejeTeje 10:54, 4 October 2006 (UTC)

Well to be honest why are you going to want to find out what an asymptote is if you're not a mathematician? It's not much use in everyday life is it? As a 16 year old A-level maths student, I found the definition of an asymptote very useful to me, although admittedly I didn't read all of the information regarding the subject because it wasn't relevant and I could glean the information that I needed from the diagrams! [unsigned comment added 09:51 19 October 2006 (UTC) by user 88.109.143.31]

Usability of an asymptote is besides the point. A wikipedia article cannot become a tutorial. At best it can give links to tutorials. So I would say that Jaysscholar is irresponsible in his/her demands. WP cannot and will never replace hard work or even a school. On the other hand, his spelling itself shows how seriously he/she considers WP. nirax (talk) 16:36, 12 August 2009 (UTC)

wrong definition
The definition of an asymptote is wrong... e.g. y=sin (e^x) has asymptote y=0, bu the curve does intersect 0 at times..

The definition was wrong; I've corrected it. Your example, though isn't right: it's not asymptotic to y=0. Perhaps you meant something like y = sin(e^x)/e^x which is nicely asymptotic to y=0 and hits it infinitely many times. It's also asymptotic to y=1! A great example! Doctormatt 03:37, 1 July 2006 (UTC)

The introductory section is wrong. Firstly: "An asymptote is a straight line or curve A to which another curve B (the one being studied) approaches as one moves along it in a positive direction." is incomplete because B can be asymptotic to A as one moves along it in a negative direction; or in both directions. Secondly: "A curve may cross its asymptote at one point, infinity, or 1/0." is wrong because some curves cross their asymptote(s) many times, perhaps without limit; also "infinity" is neither a point nor is it "1/0"; division by zero is ordinarily undefined. Note that, according to the definition I substituted and those ordinarily given in other places, if A is an asymptote of B then B is an asymptote of A; it follows that any line, ray or curve has limitless asymptotes. What is not immediately obvious is that even a line segment can have asymptotes by the definition I gave. Does this pose difficulties somewhere? Myron 05:30, 16 November 2007 (UTC)

I can't help but saying after reading the above line of conversation : 'Yeah;  math should be made easy; Wickipedia should do just that  and these explanations leave me (sic) about where I found myself when confronted by division and word problems in elementary school    In any case it does make it somewhat clearer; if doing nothing more than at least familiarizing me with some of the relevant jargon;  but yeah; I wish they'd make this geometry as simple as numbers/counting  68.153.193.155 (talk) 22:25, 27 July 2011 (UTC)

1 July 2006 changes
I just made a bunch of changes to the page. I expanded (i.e., made it longer) the layman's definition to make it both (I hope) more accurate and clearer. I really want to emphasize the point that the curve can cross the asymptote, so phrases like "never quite reaches" are not correct.

I didn't use the phrase "arbitrarily closely" since it does seem to cause problems.

I think it important to make it clear whether one is talking about asymptotes of graphs of functions, or of more general curves. To this end, I split a lot of the page into a "graphs of functions" section.

Let me know what you think.

Doctormatt 03:37, 1 July 2006 (UTC)

As a pretty keen maths student finishing high school (probably a pretty average reader of the page) I reckon the new definition is superb. *thumbs up* good job. Theonlyduffman 01:27, 1 November 2006 (UTC)


 * Still have problems. The asymptotic behavior doesn't have to occur towards infinity. You can have curves in the planes that are asymptotic but are bounded. Imagine two spirals spiraling to the origin. They are asymptotic but the distance tending to zero occurs at the origin. What it matters is that the distance tends to zero when the parameter tends to the right limit in the parameterization. Certainly this page needs to go from the very informal idea to the strict and more general definition and back again to the usual asymptotes (straight lines) that kids study in calculus.  franklin   05:52, 11 December 2009 (UTC)
 * You may have a valid issue but from what I can find, the problem isn't with the article. Wikipedia depends on existing published works to establish credibility, so unless there is published material to support the changes you have in mind they shouldn't be made to the article. I've tried to find material that goes beyond the linear asymptote definition and have tried to include what is out there in the article, but there seems to be very little to find and what there is seems to somewhat controversial. It's not up to Wikipedia to fill in gaps in the mathematical literature, so if there is a corresponding gap in an article there is little to be done.--RDBury (talk) 08:35, 11 December 2009 (UTC)


 * The typical area in which any kind of behavior is needed is in complex analysis for the study of asymptotic values in the context of Picard theorem and related topics. A book, that have a lot about that is Markushevish, Theory of the functions of complex variable. Computer scientics also need other definition besides the usual limit of the diference and definitely need not only straight lines (to measure complexity of algorithms).  franklin   09:56, 11 December 2009 (UTC)


 * This might be a language issue. Typically in English, "asymptote" refers to what might more properly be termed a linear asymptote.  (It is instructive to compare also with the Encyclopedia of Mathematics entry here.)  One can indeed talk about functions being asymptotic, but they are not usually called asymptotes (at least in the English-speaking world), as that word already seems to have other connotations.  There is indeed an area of asymptotic analysis which studies functions that are asymptotic with each other, but again the word asymptote itself is generally not used here.   Sławomir Biały  (talk) 23:47, 12 December 2009 (UTC)


 * Also, none of the three volumes of the Markushevish treatise seems to mention asymptotes (at least not in the index). Should this be removed from the list of references in the article?  Sławomir Biały  (talk) 23:56, 12 December 2009 (UTC)

Vertical Asymptote
Is it correct that x=a is an asymptote of f(x) if "limit of f(x) as x->-a = +or-infinity". It is written in the article.
 * It doesn't say that. If says x=a is an asymptote if the limit of f(x) as x->a from the left = + or - infinity, or the limit of f(x) as x->a from the right is + or - infinity.  "-a" never comes into it.  I just noticed though that the one-sided limits are indicated using superscripts, so I changed it. Doctormatt 17:37, 25 August 2006 (UTC)

Local Wobbling?
its a pretty odd description, combining the technical with the totally un-technical. Surely there would be a better way to explain that a function can become nearer to and further from a function locally so long as overall it is approaching the asymptote?
 * Can you be more specific about where in the article you wish to see improvements? Thanks. (p.s. don't forget to sign you comments with four tildes) Doctormatt 02:20, 1 November 2006 (UTC)

Page needs re-formatting
Could someone please reformat the page it is in dire need of this, the images are oversized and the text is splattered accross the screen in random places. I don't know how to otherwise i would have done it myself, thanks.
 * It's better now IMO.--Svetovid 12:40, 31 May 2007 (UTC)
 * Could still use a little work, I don't know how to either but in the last section their is  a picture covering the links.  —Preceding unsigned comment added by 24.8.206.37 (talk) 22:39, 1 October 2007 (UTC)

Error in the definition
A is OX axis, B is OY axis. For any d: $$\left\{(x,0):|x|>\frac{d}{\sqrt{2}}\right\}$$ on A, and $$\left\{(0,y):|y|>\frac{d}{\sqrt{2}}\right\}$$ on B are the sets mentioned in the current definition. So OX is an asymptote of OY??? 83.5.233.32 (talk) 12:10, 12 January 2008 (UTC)


 * OX and OY are not asymptotic. The definition is not worded great, and indeed it doesn’t actually define what is meant by the “distance from A to B” Typically one would use either the vertical distance or horizontal distance, but others are possible. In any event, the distance is not measured by the distance from the points to an intersection. Being perpendicular OX and OY would only have a finite [vertical] distance from each other at y=0 so they’re certainly not asymptotic using that distance (nor will they be with any other common measure of distance between curves).
 * A more interesting nonexample would be y1=5x and y2=3x. The vertical distance here is y1-y2=2x. Given any d, we can choose x so that 2x>d, so they’re not asymptotic (at least as x becomes large in the positive direction). GromXXVII (talk) 12:31, 12 January 2008 (UTC)


 * It’s also doesn’t make entirely clear that the points A and B do not “move” independently. For instance, given two asymptotic curves such as y1=1/x, and y2=1, given a point on y1 (say, A=(x,1/x)), I can always find a point on y2 that is as far as I like from it (say B=(b+x,1)), and the distance between A and B would be greater than whatever b I like.
 * That’s probably why we normally use vertical distance (or horizontal distance for vertical asymptotes) most of the time.
 * Another way of finding the distance (well finding the points to measure the distance I mean) might be to take some point A on the first curve, and extend a line perpendicular to the curve at the point A until it intersects the second curve, and call that intersection point point B. GromXXVII (talk) 12:47, 12 January 2008 (UTC)


 * I think, the problem is not in a distance measure (a distance between two sets A and B is well defined in mathematics as $$dist(A,B)=\inf(\{dist(a,b):a\in A, b\in B\})$$), but in the definition of the compared parts of the two curves. I think the proper definition should be something like this:
 * Curves A and B are asymptotic iff there exist continuous functions $$x_A, y_A, x_B, y_B \colon\mathbb{R}_+\to\mathbb{R}$$, such that the following conditions are all true:
 * $$(\forall t\in\mathbb{R}_+)\ (x_A(t),y_A(t))\in A$$
 * $$(\forall t\in\mathbb{R}_+)\ (x_B(t),y_B(t))\in B$$
 * $$\lim_{t\to\infty} x_A(t)=\pm\infty \vee \lim_{t\to\infty} y_A(t)=\pm\infty$$
 * $$\lim_{t\to\infty} (x_A(t)-x_B(t))=0$$
 * $$\lim_{t\to\infty} (y_A(t)-y_B(t))=0$$
 * But this is my OR. Best, Olaf m (talk) 10:11, 13 January 2008 (UTC)


 * That seems to be a nearly correct definition, although I hope not the simplest. It seems to me that the domain of the functions xa,ya,xb,yb would need to be less restrictive.
 * It doesn't matter. For any desired domain (a,b) you can always take $$t=\frac{b-a}{b-u}-1$$ to obtain $$t\in \mathbb{R_+}$$ required in the formulas above. Olaf m (talk) 18:02, 13 January 2008 (UTC)


 * Because in order to catch a vertical asymptote it seems like they would need a domain that’s bounded at the vertical asymptote: as a continuous function couldn’t go to infinity and then exist at some finite value at the asymptote. (And of course the possibility for a horizontal asymptote only where x is negative).


 * Why, let's consider A: $$y=\cot x\;$$ and vertical asymptote B: $$x=0\;$$. Then $$x_A(t)=\arccot\ t,\ y_A(t)=t,\ x_B(t)=0,\ y_B(t)=t\;$$ This definition works also for vertical asymptotes. Olaf m (talk) 18:02, 13 January 2008 (UTC)


 * As much as I don’t like the current definition in the introduction I think it’s technically correct as long as the distance measure is defined correctly. Whether or not that’s appropriate I don’t know. I’ve tried several times to rewrite that definition: because it seems like the intention is to give a definition accessible to someone that might not know much math; but I think it completely fails at this purpose.
 * Specifically I’ve tried to write the definition without using the idea of a limit, or other tools beyond basic algebra: and haven’t come up with anything that’s not overly cumbersome.
 * One problem also is that seemingly none of the basic algebra books give a good definition (they typically split it into horizontal, vertical, and in some cases oblique linear asymptotes and don’t have a real definition but only a picture and a few examples). Then the more advanced books assume the reader knows what it is and don’t seem to define it at all. I have found that elementary statistics and calculus texts sometimes give a workable definition, but barely better than the algebra books and generally involving limits. I haven’t looked at any statistics book intended for use without calculus, such might have a good definition: but probably only for horizontal asymptotes which isn’t very useful.


 * The current definition in the article is not correct, even if the distance is defined correctly. If you define the distance between sets as the Euclidean distance between their nearest points, then the example above (with asymptotic OX and OY) is a proof, that the definition is wrong. $$\left\{(x,0):|x|>\frac{d}{\sqrt{2}}\right\}$$ forms "points on A beyond which the distance from A to B never exceeds d" and set $$\left\{(0,y):|y|>\frac{d}{\sqrt{2}}\right\}$$ forms "points on B beyond which the distance from A to B never exceeds d". Olaf m (talk) 18:02, 13 January 2008 (UTC)


 * I still think the definition is correct if you use a workable measure of distance. The counterexample works for the measure of distance you gave, but not in general. Try for instance using sup instead of inf as the measure of distance and it should work.
 * In any event, the current definition is not very good. GromXXVII (talk) 19:33, 13 January 2008 (UTC)


 * Supremum is also bad. With supremum no two curves will be asymptotic, even if identical. You should take a measure like $$\sup(\{\inf(\{dist(a,b):b\in B\}):a\in A\})$$ to achieve the goal, that is find minimal distance for any given point in A, and then take maximum from this set. Sounds a bit complicated. Olaf m (talk) 20:49, 13 January 2008 (UTC)


 * I had also come up with something like this: to define two functions to be linearly asymptotic if for arbitrary epsilon, beyond some point on both curves an epsilon-tube can be placed around them. That would only be feasible though if there’s a wiki article that talks about epsilon-tubes which I couldn’t seem to find (though there probably is, but by a different name). It also leaves functions asymptotic to something other than a line in the cold. GromXXVII (talk) 13:13, 13 January 2008 (UTC)


 * What about $$(x,\cot x)\;$$ and $$x=0\;$$? How can one place epsilon tube around the whole plot of cotangent? What does it mean "beyond some point", if a curve is for example $$\{(x,y):x\in\mathbb{Z}\vee y\in\mathbb{Z}\}$$ ? Olaf m (talk) 18:02, 13 January 2008 (UTC)


 * Hmm, well one couldn’t put an epsilon tube around the whole plot of most functions. But you could put one around the piece with the linear asymptote your interested in. In the cot case, for sufficiently large (or small) y-values. So for instance one such tube might be plotted as {(x,y)|x=d, or x=-d, y>cot(d/2)} GromXXVII (talk) 19:33, 13 January 2008 (UTC)


 * Yes, but in that case you have to define what piece of the plot in general can be taken, which is not obvious. Olaf m (talk) 20:49, 13 January 2008 (UTC)


 * I did some more searching, I found this which is close, and might be modifiable to a function instead of a value. In Real and Complex Analysis Rudin defines an asymptotic value of an entire function as &alpha; when there is a continuous mapping &gamma; of [0,1) into the complex plane such that &gamma;(t)&rarr;&infin; and f(&gamma;(t))&rarr;&alpha; as t&rarr;1. GromXXVII (talk) 13:53, 13 January 2008 (UTC)
 * Yes, it is possible, and you will get exactly my definition stated above, but expressed in terms of the complex analysis:
 * Let A and B be curves expressed as subsets of the complex plane.
 * $$a:[0,1)\to A,\ b:[0,1)\to B$$
 * $$t\to 1\Rightarrow a(t),b(t)\to \infty\wedge a(t)-b(t)\to 0$$
 * A bit shorter, even if not more understandable. ;-) Olaf m (talk) 18:02, 13 January 2008 (UTC)

What about something like this:
 * A and B are asymptotic iff for any positive $$\epsilon$$, there exist their unbounded subsets $$A^\prime\subseteq A$$ and $$B^\prime\subseteq B$$, such that distance between any point in $$A^\prime$$ and the nearest point in $$B^\prime$$ is lower then $$\epsilon$$.

I think, this form of your definition is both proper, and clear. What do you think about it? Olaf m (talk) 20:49, 13 January 2008 (UTC)


 * I think that is quite good. I’ll go make some appropriate changes. GromXXVII (talk) 22:33, 13 January 2008 (UTC)


 * Well, I was going to change the intro to
 * A curve A is said to be an asymptote of curve B when the following is true:
 * For any positive $$\epsilon$$, there exists unbounded subsets (pieces of the respective curves) $$A^\prime\subseteq A$$ and $$B^\prime\subseteq B$$, such that the distance between any point in $$A^\prime$$ and the nearest point in $$B^\prime$$ is lower than $$\epsilon$$.


 * In other words, as one moves along B in some direction, the distance between it and the asymptote A eventually becomes smaller than any distance that one may specify.


 * If a curve A has the curve B as an asymptote, one says that A is asymptotic to B. Similarly B is asymptotic to A, so A and B are called asymptotic.
 * But then I realized I think something's not quite right. Take y1=sin(x), y2=0 for instance. A' could be {(x,y)| x=n&pi;, y=0}. What about requiring A' to be connected? As long as the curves have a finite number of discontinuities that should work; but I'm not sure about the other cases.
 * Still going to add the formal definition though, it could use that. GromXXVII (talk) 23:05, 13 January 2008 (UTC)
 * Well, you are right. I think, you should require both A' and B' to be connected, or the problem persists. But it is still not enough. For example a curve $$\{(x,y):y=\tfrac{1}{x} \wedge \sin\tfrac{1}{x}>0\}$$ according to the standard "functional" definition has an asymptote x=0, because $$\lim_{x\to 0^+} y(x)=\infty$$, but you cannot find any connected unbounded subset of it at all. I have no idea, how to fix it, I'm not even quite sure if it needs beeing fixed. Rudin's "asymptotic value" also requires a connected unbounded curve to be a subset of domain of f. The formal definition you have been just added (that is my OR) requires the same thing. Olaf m (talk) 23:42, 13 January 2008 (UTC)


 * What’s the standard “functional” definition you’re referring to? I’m not familiar with how to treat that curve if it indeed has an asymptote. The typical elementary definition with a limit to infinity or negative infinity typically requires the function to be continuous in the reals on some open interval near the point, so that can be applied here unless if there is a more general definition.


 * I added that definition because it adds to the page, and there isn’t a current better alternative. The fact that Rudin’s asymptotic value is so similar, and that I haven’t been able to find a published definition for asymptote or asymptotic makes it not seem like a problem. In as far as I can tell whatever the definition on this page is, is likely to be along the same lines because no reliable sources have been found [to my knowledge] with a true, rigorous definition: but yet the definition of a well known term certainly is not original, only the presentation of it.


 * For now, I’ll add this in the case where the curves A and B are connected to begin with. Not completely general, but better than the current intro because it’s correct only by the fact that it’s so vague with it’s use of undefined terms it’s difficult to make sense of.GromXXVII (talk) 01:27, 14 January 2008 (UTC)

The main problem with all these definitions is not that they are incorrect, it's that they're OR. Provide a reference with the definition and then it will be correct. Another big problem is that they are not accessible to the target audience. This article should be understandable to freshman level calculus students and all this talk about locally connected unbounded sets will be over their heads. It's not necessary nor desirable to present a fully rigorous and general definition in the lead section; this in an encyclopedia, not a graduate text. Better to put a non-rigorous, non-general but understandable definition in the lead and put the mathie version it it's own section.--RDBury (talk) 16:44, 2 August 2008 (UTC)

Slant Asymptote
The section on Slant Asymptote is rather confusing. My math teachers are rather confused looking at it. Can someone who knows a little more about the slant asymptotes both check the math and make it a little easier to understand. --Omnipotence407 (talk) 14:13, 16 January 2008 (UTC)

Much confusion.-- ( fi ) 23:24, 4 March 2008 (UTC)

The Horizontal Asymptote is rathe confusing.
I personally cannot make head or tail of said section. I'm currently doing alegebra homework, and what my book has looks nothing like what I'm seeing there. I'm not saying "delete" it, I was just wondering if someone could make it a little simpler to understand. Thanks! Paladin Hammer (talk) 05:29, 18 April 2008 (UTC)

Intersecting asymptotes? Really?
Certainly a curve can intersect its asymptote, even infinitely many times, within some finite range. But are there really definitions that allow a curve to continue to intersect its asymptote repeatedly as it approaches infinity? The word literally means "not meeting." My teachers used it pretty consistently to mean not just any old limit at infinity but one that was approached one-sidedly. Not R (talk) 04:38, 24 July 2008 (UTC)
 * It’s hard to find a definition for a general asymptote – the closest I’ve seen in a textbook was in terms of complex analysis. However, the simplified definition used in most calculus texts does allow a repeated intersection as it approaches infinity. For instance, in Stewart the definition is
 * “The line $$y=L$$ is called a horizontal asymptote of the curve $$y=f(x)$$ if either $$\lim_{x\to \infty} f(x) = L \mbox{ or } \lim_{x\to -\infty} f(x) = L$$”
 * This indeed allows something such as $$sin(x)/x$$ to have asymptote $$y=0$$ . Also note however that for a rational function this is an impossible case because rational functions have finitely many turning points and discontinuities. This is significant because many algebra texts introduce the concept of an asymptote in the special case of a rational functions before giving a more general introduction. GromXXVII (talk) 10:55, 24 July 2008 (UTC)
 * By that definition, any line is an asymptote of itself! Not R (talk) 17:17, 9 August 2008 (UTC)
 * What about the following function:
 * $$f(x) =

\begin{cases} 1-|x|, &  -1<x<1\\ 0 & \mbox{elsewhere} \end{cases}$$
 * Does it have two horizontal asymptotes at x=0? --Slashme (talk) 12:31, 11 August 2008 (UTC)
 * A function can have horizontal asymptotes at $$\infty$$ and $$-\infty$$ and vertical asymptotes at other points. I don't know what a "horizontal asymptote at x=0" would be. &mdash; Carl (CBM · talk) 14:50, 11 August 2008 (UTC)
 * I think he means f(x)=0, in which case the answer is "yes". --Tango (talk) 21:21, 11 August 2008 (UTC)
 * Oops, yes, that's exactly what I meant! --Slashme (talk) 06:44, 12 August 2008 (UTC)

It looks like we should do a literature survey. Here's a start; please fill in more. &mdash; Carl (CBM · talk)


 * Books that define a horizontal asymptote only in terms of the limit value, with no restrictions on intersections:
 * Thomas and Finney, 9th ed., p. 224
 * Edwards and Penny, 4th ed., p. 245
 * Stewart (details needed)
 * Books the restrict the number of intersections of a function and its horizontal asymptote:
 * Please add any you find
 * Please add any you find


 * I can add some textbooks to the first pile. Calculus with Applications 8th edition (Lial, Greenwell, and Ritchley; pp 81-83, 150, 314, 315) which applies the concept almost exclusively to rational functions, but defines it in terms of limits. Calculus, 8e (Larson, Hostetler, Edwards; pp 84, 85, 199, 211, 701, A6) which also defines them in terms of limits, and applies them to somewhat more general functions. College Algebra Fifth Edition (Sullivan; 345-351, 522-524) which defines them in terms of limits, but paradoxically also says, "an asymptote is a line that a certain part of the graph of a function gets closer and closer to but never touches (p 346)." It applies them to rational functions and hyperbolas. I'm also holding some computer science textbooks that define asymptotic behavior in a very different way, one that's probably not relevant to this article. My analysis textbook skips the word entirely and just uses limits. I would like to point out that vertical asymptotes, horizontal asymptotes, slant asymptotes, and more general asymptotes (I list them in the order they are usually introduced) are extremely different. There's a reason they're discussed separately, even to the point of being introduced in different chapters of the book in some cases. There's no reason to try to cram them all into one definition on the first go-round. As I see it, there are three goals that people have stated so far: That the article be rigorous, or at least contain a rigorous section, that it be accessible to people who don't know the subject (asking that it be accessible to people who don't want to is going a bit far), or at least contain a section near the beginning that's accessible, and that it cover every definition of the word people might come across, or at least every definition in an introductory calculus textbook. I don't see any reason these can't all be accomodated. In fact, if anyone can find a free-access calculus textbook online, I don't see why we can't rip the exposition straight from there. It's bound to wind up almost identical to everyone else's. Black Carrot (talk) 02:39, 12 August 2008 (UTC)


 * Two further points. First, there's nothing wrong with a line being asymptotic to itself. It's an uninteresting case, what people in the biz call "trivial", but there's nothing wrong with that. Second, there's nothing wrong with a word having different definitions in different contexts. I'd consider a thorough article on functions remiss if it never mentioned "partial" or "multivalued" functions, for which the modifier is often ignored. Similarly, the word "graph" in graph theory can mean a dozen different things, as can the word "ring", depending on who's using it and how willing they are to repeat things like "with unity" every other sentence. In some contexts, the only asymptotes you're interested in are the ones where the curve eventually stops intersecting it, and then that would be a reasonable definition. The geometric definition of tangent is the same way. In calculus though, you eventually have to replace those definitions with more flexible definitions in terms of limits. When you make that transition is up to your teacher, but it's usually made pretty quickly. Black Carrot (talk) 02:49, 12 August 2008 (UTC)
 * It's not that it's trivial, it's that it's contrary to the literal meaning of the word "asymptotic" (not meeting). "Tangent" (touching) is also trivial in the case of a line, but sensible:  the line does touch itself at every point. Not R (talk) 02:54, 19 August 2008 (UTC)
 * Words are often used in ways that are different than their literal meaning (especially if the literal meaning is only understood in another language). In this case, it appears from the textbook survey that any sense that a function cannot touch its asymptote has been lost to mathematical generalization. &mdash; Carl (CBM · talk) 12:22, 19 August 2008 (UTC)

Cuz Why Not
Since people are suggesting we strive for ridiculous amounts of generality and definitions so advanced that the people suggesting them don't know what they mean, I figured I'd throw a little more wood on the fire. Why not introduce non-archimedean definitions of asymptote while we're at it? We could mention the several geometries, such as nonstandard analysis, where it's perfectly reasonable to say that a curve is asymptotic to a line if, at infinite distance from the origin, the deviation of the one from the other is infinitesimal. Or we could point out projective geometry, coordinatized by the projective reals, where a line is asymptotic to a curve precisely when it's tangent to that curve at infinity. That viewpoint even gives a nice visual justification for allowing them to intersect each other a lot. What do you think? Black Carrot (talk) 06:58, 12 August 2008 (UTC)

Error term?
If the degree of the numerator is more than 1 larger than the degree of the denominator, there will generally still be an error term that goes to zero as x increases, but the quotient will not be linear, and the function does not have an oblique asymptote.

What is an error term? Is that a real phrase used by mathematicians? It looks like it is talking about the remainder. If that is a phrase in common use it seems unfortunate. There is nothing about a remainder that is an "error" unless you messed the division algorithm up. So if this isn't a very commonly used phrase I'd vote to rephrase that--and the section isn't written very clearly to boot. And the word "generally" seems to be entirely misused here. Its anything but general, the contributor seems to have been talking about the inverse of generally: as there isn't always a remainder.

The more I read this section the more I think it should be completely rewritten. [User:Brentt|Brentt]] (talk) 20:51, 16 December 2008 (UTC)
 * An error term, the way the contributor (me, I think) was using it would be the difference of a function and an approximation. In this case the quotient is being used as an approximation for the dividend. I’d say remainder is a better term to use here because it is more precise.
 * I may have been using “generally” too colloquially – I hear it used a lot to mean that something will usually happen, although leaving the possibility that in a special case it might become trivial, but not all the time. Here it becomes trivial when the denominator divides the numerator. GromXXVII (talk) 14:56, 17 December 2008 (UTC)

refimprove tag
There are no references listed for this article. As noted above, some of the definitions may be OR.--RDBury (talk) 11:46, 14 April 2009 (UTC)
 * I added a reference and replaced the definition in the article with the definition used there. I did have to update the language somewhat but I didn't change the meaning. I'm leaving the refimprove tag as the article still needs more references, preferably more modern ones.

Asymptotic
An ASymptot is a line or curve or set of poits that a curve or line aprroaches but never gets to touch them, though the horizontal asymptotes can be closed.--Romutujju (talk) 11:48, 5 December 2009 (UTC) I removed the definition of asymptotic since it's not the same (as far as I can tell) as an asymptote. For example, x and x+1 are asymptotic in the sense that the ratio approaches 1 but they are not asymptotes of one another.--RDBury (talk) 12:43, 8 July 2009 (UTC)
 * What you removed is not what you were compaining complaining about. There is yet another definition of asymptotic which might apply, the existing one is correct for curves being asymptotic, if not for functions.  — Arthur Rubin  (talk) 13:55, 8 July 2009 (UTC)
 * This is why this article desperately needs some external references, some people are using one definition, some people are using a different one, and some people are using one they made up and no one else has ever heard of. Can you cite a source for your definition of asymptotic curves? I'm trying to use the definition in Fowler until someone finds a better source, but Fowler's definition is not symmetric so it only makes sense to speak of one curve being an asymptote of another, not two asymptotic curves. I'll leave the sentence in there but I'm adding a fact tag.--RDBury (talk) 17:49, 8 July 2009 (UTC)
 * Also, Mathworld is a good source of made-up words. Removing those.  — Arthur Rubin  (talk) 19:14, 8 July 2009 (UTC)
 * The definitions here for curves being asymptotic is clearly symmetric, and the most detailed formal definition at the end of the #Definition section is symmetric. The multiple definitions for functions being asymptotic (although I don't see any as standard) are symmetric, but not all obviously so.  — Arthur Rubin  (talk) 19:31, 8 July 2009 (UTC)
 * I'll leave the fact tag in place, because I hadn't seen the definition before, but it's the text of the article. It should be tagged there, instead of in the lede.

Formal definition - just an OR
The "formal definition" in this article is just my Original Research, taken from the discussion above. I did't expect it to be inserted in the article. I think, it should be removed and replaced by the proper one, referenced in the scientific bibliography. Olaf (talk) 17:16, 7 August 2009 (UTC)


 * I added it to the Pages Needing Attention list for good measure. The problem is that the scientific literature doesn't seem to have a lot of coverage on this and what is there is usually incomplete or inconsistent. Every college freshman calculus text defines horizontal and vertical asymptotes of functions and most cover asymptotes of hyperbolas, but none give a really satisfactory general definition because that would be outside the scope of a freshman calculus class. I've gone to the usual math web resources and have found little useful. MathWorld just has a picture and a vague indication of the concept rather than a definition, Planet Math is better but only defines linear asymptotes, MacTutor also only defines linear asymptote and their definition is different from the one in Planet Math, and Springer Enc. of Math doesn't go much farther than the freshman calculus definition. I added the Fowler reference because it had the most general definition I could find on the web, but it's not recent and the definition isn't symmetric. (For example y=sin(x2) is an asymptote of y=0 under this definition but the reverse is not true.)--RDBury (talk) 20:19, 7 August 2009 (UTC)
 * Well, if nobody in the scientific world defines nonlinear asymptotes, the Wikipedia article should mention only the linear ones as well, shouldn't it? Otherwise, how could you prove, the described object is still called an asymptote, if nobody calls it in this way? Or, if it is somehow defined in Fowler, use his definition, however it must be very odd... y=sin(x2) as an asymptote of y=0 doesn't make sense to me. Could you cite that definition? Olaf (talk) 05:35, 9 August 2009 (UTC)


 * I agree we should look for a definition we can cite, so I removed your definition in the meantime. There is a link in the article to a PDF of Fowler's book, where you can find his definition on p. 89. But it is written in archaic language. The reason it is not symmetric is that it defines what it means for the graph of a function y = f(x) to be asymptotic to an implicity defined curve g(x,y) = 0, but does not define what it would mean for two implicitly defined curves to be asymptotic. &mdash; Carl (CBM · talk) 12:05, 9 August 2009 (UTC)
 * Well, Fowler's definition is quite nice, and it doesn't imply, that $$y=\sin(x^2)$$ is an asymptote of y=0: if a point P on the curve y=0 tends to infinity, distance from $$y=\sin(x^2)$$ tends to infinity as well... I believe, if he had written "the curve f(x,y)=0" instead of "the curve y=f(x)" the definition would be equivalent to mine. Best, Olaf (talk) 21:37, 12 August 2009 (UTC)


 * I'm starting to come around to Olaf's point of view with regard to non-linear asymptotes. Linear asymptotes are most often used in the literature I've seen and using projective geometry they can be seen as a generalization of tangents. Definitions for non-linear asymptotes may be found, but none of them seem to have become common usage in a geometric context. Non-linear asymptotic behavior of functions should be left in the Asymptotic analysis article.--RDBury (talk) 18:02, 25 October 2009 (UTC)

Asymptoting
New page? Asymptote as a verb. This is widely used as a verb- two examples follow: http://adrianmonck.com/2007/09/asymptoting-towards-zero/ and http://www.springerlink.com/content/07l2250t7p6l5411/ Asymptote was first used as a verb by Rich Tighe, who was overheard to say, "Who has been asymptoting the pie?" Electricmic (talk) 19:46, 15 November 2009 (UTC) I want to concede that my claim for my friend as the source of "asymptoting" as a verb was unsourced and appreciate the well sourced revert of my assertion of first use. Is there still a way to include the fun use of "Who has been asymptoting the pie?" as an example of how math talk has entered common speech?

Recent changes
I don't believe the recent changes made to this article are an improvement. First, the definition that was given before was from a reliable source, but the new definition is not. The new definition seems to based on a confusion between an asymptote and asymptotic analysis which, though related, are different concepts, the first being geometrical and the second being analytical. Second, the new material only covers curves of the form y=f(x), so, for example, the article does not cover asymptotes of the hyperbola x2-y2=1. Third, the definition for metric spaces seems to be pure OR. The article has been a haven for OR definitions in the past and this represents a step in the wrong direction. The harm is not the great because the article was not in very good shape to begin with, but changes such as these to a highly visible article should be discussed beforehand.--RDBury (talk) 22:48, 12 December 2009 (UTC)


 * Very little is gained with the style of attempting to formulate the most general possible definition up front. A much better structure for this kind of article (also bearing in mind the likely readership) is to define an asymptote first in the usual way: as a (linear) asymptote of a curve.  (See, for instance, Pogorelov's classic textbook on differential geometry.)  More general definitions, preferably with sources, can then be presented in a "Generalizations" section.  This is a problem that both the new and old versions of the article seem to suffer from, so it is by no means a criticism limited to the recent revision.  But the problem seems to have grown.   Sławomir Biały  (talk) 23:34, 12 December 2009 (UTC)


 * I agree. It seems to be a re-write to no good purpose which has made it less clear mathematically without really adding anything. In addition to the above points it reads in parts more like an essay, trying to an argue points, and from the start has problems with an introductory paragraph that's too long. The formatting too is all over the place with an untidy mix of formats for the inline math, poor sentence and paragraph structure. --JohnBlackburne (talk) 00:39, 13 December 2009 (UTC)


 * I beg your pardon. In the previous version there were even mistakes and inconsistencies. It is true that that it need some work on the edition of formulas. Again, if you have an idea, go ahead and do it. The introductory paragraph is not that long, it looks long because of the two big formulas, but thats a format problem, it can be changed. I have no strong feeling about the actual definition of asymptote but the ideas in the introductory paragraph are quite important to let understand what the nature of the concept and its purpose.  franklin   01:38, 13 December 2009 (UTC)


 * The introductory paragraph of today's featured article is only one line less that this one.  franklin   01:51, 13 December 2009 (UTC)


 * The definition that was given before was using curves also. look here. What I did was to take all the mess that was there and write it in a correct way. With consistent notation and structure, and intension. Also, the new structure of the article is done in such a way that if needed the definition can be changed easily. A stronger and consistent emphasis is made on the use of parameterization since, although many times avoided by introductory courses, it is quite important to understand the concept. You said an asymptote is a geometric concept. Well, let me show you that it is in fact closer to being analytic (at least the definition closer to the taught at school). Take your hyperbola (but have in mind that we need also any other implicit curves), in order to define the asymptote you need a parameterization. Now, if you use different parameterization a line can be asymptote and for some others it wont. Now, a geometrical meaning, in which there is no dependence of the parameterization, can be given, and comes essentially from the same informal first paragraph. Now, if the formal definition is going to be geometrical (no parameterization) then many more eyes will go wide open since is going to look very different to what is done at school or in most books. Now, assuming the need for parameterizations.it comes the question of why to put only parameterizations on the curve and also on the line? In any case, be bold, if you think you have and idea of how to improve an article go ahead and do it. What I did was just to take what was there, and without changing the content, improve the presentation.  franklin   01:31, 13 December 2009 (UTC)
 * The earlier parts of the article should be understandable to as wide an audience as possible (see WP:MTAA and WP:MOSMATH). Since this is a topic that is covered in most high school level mathematics curricula, discussing metric spaces early on in the article is totally inappropriate.  First and foremost, the article needs to shift focus more onto linear asymptotes.  Other asymptotes are considered very rarely in the available literature, and should be covered deep down in the article, not right in the first section.  Secondly, the current lead section, rather than waffling about nonsense like "qualitative" versus "quantitative" properties, should state briefly what a horizontal asymptote, vertical asymptote, and oblique asymptote are.  This is the kind of information that would be helpful to readers, not confusing pseudo-pedagogical prose.  Thirdly, while the former incarnation of the article also attempted to give a fully general notion of asymptote early on, at least it gave a much better plain language description of it.  The new "General definition" section is totally incoherent, engaging in what appears to be more pseudo-pedagogy, and then in the end even fails to present a single accurate definition at all.  The "parametric curve" definition is wrong&mdash;in the sense that it is not in agreement with sources, nor with the old version of the article.  Anyway, as I've already said, the general definition should be way down in the article, not right in the first section.  Finally, and I cannot emphasize this enough, it is very important to give sources (and, indeed, in some cases to cite them), and for the text to agree with those sources, and to represent all views according to the weight they receive in reliable sources.  In particular, since most sources do confine attention to the linear case, we should place the bulk of the emphasis on this case as well.  Sławomir Biały  (talk) 02:36, 13 December 2009 (UTC)


 * Do, do all of that. About the pseudo-pedagogy and the waffling. Understanding the importance of that waffling is at the very center of the notion of asymptote as approximation, it distinguishes it from other approximation notions like, order of growth for instance, or limiting direction. Also that understanding leads to the evolution of the concept to the notion of asymptotic, which is so connected to that of asymptote that led many to treat them as the same thing. Of course I understand that it takes more than being just a mathematician to grasp all of that. But c'es la vie.  franklin   03:29, 13 December 2009 (UTC)


 * Given the amount of damage that's been done to the article the best approach would be to put it back where it was before this major overhaul was attempted. Then if you still want to 'fix' it there would be two ways to do it. Either do it incrementally, fixing those things that you think have problems to slowly move towards an improved version. That will make it easier for others to challenge your changes individually and fix them. Or attempt a re-write in user space which others can give feedback on before it's merged with this version. Quibbling over odd lines when the whole thing needs re-working will take forever. --JohnBlackburne (talk) 09:40, 13 December 2009 (UTC)


 * Amount of damage! I can only laugh. Tell me, mathematician, tell me what is the damage. Have you read the article at all? Look, the difference from what the article had and my first big edition is almost null. The edition consisted mostly in taking what was there and structure it and give it a consistent language to the point of the concept of asymptote. For this I removed errors, repetitions, explained things that in the talk page were topic of large discussion and confusion. Remember when in school, when you studied mathematics they taught you how to give a good proof of an statement? Well, if you are going to say something as above try to follow what you are suppose to know. Many times in the process of proving some intuitive fact one arrive to its falsehood. An read the article.  franklin   14:00, 13 December 2009 (UTC)
 * As I don't have all day I'll limit my comments to the lead section. Although someone has started fixing the errors it still has the section on entomology which is wholly unnecessary. To introduce something that "can lead to confusion" in the lead section is of help to no-one, especially as it seems to be a minor, historical confusion, i.e. not one that would trouble someone studying the subject today.
 * More generally it worrying when all the references are so old. Although this mathematics is at least as old as those sources, so they should be comprehensive and correct, terminology and common practice evolves down the years so books written a hundred years ago might cover the subject in a very different way to modern sources. --JohnBlackburne (talk) 14:26, 13 December 2009 (UTC)


 * Well, the amount of damage seems to be difficult to explain isn't it? About the etymology of the word. It is a big source of confusion and is not a mere historical thing. Many schools (not well informed) keep teaching asymptotes like that. You can see also part of that big problem in this talk page. Many posts are related to that issue. If you find old literature, is because most of them is supporting the statements about old conventions. The main serious use of asymptotes ( as only lines) is done in the curve tracing business. But PC have moved that away from the interests of this days. Then you will see it used only in the basic text books and in little areas as descriptive geometry, curve tracing, architecture...  —Preceding unsigned comment added by Franklin.vp (talk • contribs) 14:35, 13 December 2009 (UTC)


 * Also, all that waffling is explaining why is not longer in the concept the not-touching thing. It is indeed very choking that while the term means literally "not touching" that is not part of the concept nowadays. It is important because although this is the case, many not-so-well acquainted teachers keep teaching that. Pedagogically and and also pragmatically (even if you don't understand it or not) the purpose of a concept is much more important than the concept it self. It gives idea beyond a particular wording, gives idea of the use and also of its possible evolution.  franklin   03:41, 13 December 2009 (UTC)


 * My standard retort to a post like this is, if something is not understandable to an expert on the subject, then what are the chances it will be understandable to a non-expert reader of the article? Well, let me take the pressure off: I do understand the point being made.  But I think it only adds unneeded confusion to the article.  The paragraph has been moved to a better place, and I see that a reference (to a French source) has been added. Once again, as I have already pointed out, I think there is a language issue over the precise meaning of the word "asymptote" in English.  In French, the term asymptotique includes both the English asymptote (which is the geometrical notion) and the adjective asymptotic (which is the subject of asymptotic analysis).  I am removing the paragraph as it mixes these two notions in a way that might potentially lead to confusion.   Sławomir Biały  (talk) 13:39, 13 December 2009 (UTC)


 * You simply removed the paragraph. That you don't understand it is not proof that it is not important. You are trying to suggest that in foreign languages there is a difference to the notion of asymptote to the one in English. Such a thing have not been established. As far as I can tell, also in French and in Spanish, asymptotes are called to the usual lines, in some cases (as in English) to curves in general. On the other hand, that reference was not there for the sake of the definition but as a reference to the paragraph you removed. In that reference it is stated explicitly the importance of asymptotes as quantitative measures of the function. This fact is important is important to understand why asymptotes don't care anymore of intersecting the curve many times, is important to understand why one can pass (and do pass) to consider general curves as asymptotes, is important to understand the difference of asymptote and asymptotic growth (as used in computer science). For this reason I will revert you deletion. If you want, if you feel is needed rephrase it (which is probably needed as my English is far from being good), but that point should be made explicit in the article for the reasons above, or more blindly, because it is done in the literature.  franklin   14:14, 13 December 2009 (UTC)


 * Note: using the definition in Pogorelov gives a geometric meaning (not depending on the parameterization). Of course, it has the drawback that arctan(lon(x)) is not going to have -pi/2 as asymptote anymore. Bye-bye asymptotes in the finite plane. I like the approach anyway and I will adapt the article to it.  franklin   02:55, 13 December 2009 (UTC)

First sentence
As far as I can tell, the current first sentence of the article is meaningless:
 * "In geometry, an asymptote of a curve is another curve who's behavior is similar in some sense when traveled, simultaneously, in a given way."

If I have no idea what this sentence is trying to say, I doubt someone who doesn't know what an asymptote is will do better. &mdash; Carl (CBM · talk) 03:06, 13 December 2009 (UTC)
 * How should it be? Don't wait, change it! It is easier just to whine.  franklin   03:43, 13 December 2009 (UTC)
 * Perhaps you should look into editors' backgrounds more closely. In any case, I posted here instead of changing it for two reasons, and I apologize for being blunt in listing them. At the time I thought the best fix would have been simply to revert your changes, but I avoided doing that in an attempt to be collaborative and polite. So I posted a note here instead, so that you could fix the sentence to say whatever you were trying to say originally. If you would prefer, I can simply revert things like this in the future. &mdash; Carl (CBM · talk) 14:27, 13 December 2009 (UTC)

The true value of a mathematician
I started searching and there are old versions as far as two years ago in which asymptote was defined as a curve. It takes only a hard work that lasted for a night organizing and cleaning the mess let accumulated for a long time to awake the pride of those that were supposed to do something but didn't do it. Shame on you. Now, the useless and bad version I did is changed to use the definition of asymptote as a line and it was so easy to do. Just a tweak!  franklin  03:54, 13 December 2009 (UTC)
 * You're right in that there is a lot of blame to go around here. The article was allowed to grow very long and detailed with no hint of a reference to show that anything in it wasn't just made up. If you look at some of the other threads in this page you can see that there have been complaints about it for a while now. I, for one, was attempting to fix some of the issues and to do research to back up the notion of non-linear asymptotes with with reliable sources. In fact I was the one who added the Fowler reference you list below in the first place and I've seen the Frost book as well. The problem with the definitions given there is that they do not match each other or any other definition I could find, and the vast majority of the sources I found, including the most prominent math websites which I listed above in another thread, mention only linear asymptotes. This may be an issue with my skills at doing research or it may be a criticism the current state of the literature. There is no value in a definition unless there is a common consensus among the group who use the term as to what it means. In this case, the Fowler and Frost books give two opinions as to what the meaning should be, but the term has been use for hundreds of years by a great many people and the opinions of two authors do not constitute a consensus. I was hoping that someone with better research skills than I would be able to find such a consensus in the literature, and the article has been tagged with refimprove for a long time now with little change until the last few days. Perhaps you were also right in that some kind of radical shake up was needed to get this article out of the state of stagnation it was in. I do ask however that radical shake ups be proposed and discussed beforehand. Any edit made to Wikipedia is subject to consensus since any other editor can easily undo the change. The larger and more radical the change, the more effort should be put into building a consensus for that change. In the end, the quality of the article is the only issue that matters, and consensus building, though often slow and painful, usually results in changes that better improve the quality of an article than when this process is skipped.--RDBury (talk) 10:31, 13 December 2009 (UTC)


 * Every line is a curve. On the other hand, it is possible for one non-linear curve to be asymptotic to another non-linear curve. So I really don't see what point you are arguing. &mdash; Carl (CBM · talk) 14:44, 13 December 2009 (UTC)


 * Really? Never thought about that. How enlightening. The point is, that defining asymptotes as general curves was there there and I kept it but is largely unwanted. As you can see form the posts above. In any case the article is changing now that I did so many "step in the wrong direction" and after being for so long in a poor condition. My point was to notice how mathematicians are moved more for the sake of pride and despise for the others than for the true sake of doing something good.  franklin   14:54, 13 December 2009 (UTC)


 * If mathematicians are full of spite for others, I don't think rubbing it in their faces is likely to be a useful technique in working with them. &mdash; Carl (CBM · talk) 15:07, 13 December 2009 (UTC)


 * It is a teaser. Now they are working instead of crying and complaining. Complaining about a work that took me a whole night, that cleaned all the mess was there, and that added consistent use of terminology and clarified many delicate topics. —Preceding unsigned comment added by Franklin.vp (talk • contribs) 15:20, 13 December 2009 (UTC)

Wait a sec. Curvilinear asymptotes
References:


 * 1) Fowler, R. H. The elementary differential geometry of plane curves Cambridge, University Press, 1920, pp 89ff.(online at archive.org) (the given in the article)


 * 1) Frost, P. An elementary treatise on curve tracing

French language source "asymptotique" versus "asymptote"
I have removed the following paragraph:

"An asymptote is a quantitative way of describing the limit behavior of a curve. While you can say that $lim_{x\rightarrow+\infty}(x^2+x+1)/x=+\infty$, which can be considered qualitative information for the function $f(x)=(x^2+x+1)/x$ by describing the nature of its behavior as tending to $+\infty$, it can be said that $y=x+1$ is an asymptote of $f(x)$, which says that their limits are the same and they approach it in a similar way in some sense. Then the simpler curve $y=x+1$ is giving qualitative information as for example:


 * f tends to infinity as fast as x+1 but not as fast as x^2 and faster than $\log(x)$ (see asymptotic analysis)


 * the more precise information that f while tends to infinity as fast as x+2, in the limit, the values are more smaller than those of this function. Collectif, Dictionnaire des mathematiques: Algebre, Analyse, Geometrie, Paris, 1997 (in French)< /ref >"

Quite apart from the questionable writing here, the reference in question appears to be about a more general concept than the one treated here. See multiple posts above in which I point out that the English-language usage is somewhat different from (for instance) the French. At least one editor here seems to think that asymptote means the same thing as asymptotic analysis, and has repeatedly tried to insert such content into the article. Anyway, for a topic as well-covered as asymptotes, one can only wonder why English-language sources are not being used, despite my remarks above. Sławomir Biały (talk) 14:50, 13 December 2009 (UTC)


 * Well, I've now been reverted twice by User:Franklin.vp&mdash;despite being repeatedly told to improve the article rather than discussing complaining about it.  Sławomir Biały  (talk) 14:54, 13 December 2009 (UTC)


 * Don't cry. No one is hitting you. Notice that I haven't touched any of your changes that are for good (and not because I haven't seen them). OK, I will find you a book in English saying that asymptotes give quantitative info.  franklin   15:02, 13 December 2009 (UTC)


 * (1) It is a bit rude to say that someone is crying. (2) It is ironic that you have appointed yourself the arbiter of what is "good" and what isn't.  See the above threads on this very page.  Sławomir Biały  (talk) 15:05, 13 December 2009 (UTC)


 * When you invite others to fix the mess here, it is probably best not to continually undermine their efforts at improvement. Sławomir Biały  (talk) 15:31, 13 December 2009 (UTC)


 * Have I undermined? every one of your changes that comply with the good writing and the better understanding of the subject I have preserved. Sorry for using the word "cry" my knowledge of English is limited and I can't distinguish between subtle differences in meaning. Maybe "lament" would be better? Again, my apologies if that was rude. I have not appointed myself as arbiter, Wikipedia have appointed us all as arbiters. When I said I was just to emphasize that I will take care of my part of that appointment.  franklin   15:39, 13 December 2009 (UTC)

Pasted from my user talk
I have moved the following thread from my talk page. Please discuss changes related to the article here, not there. Sławomir Biały (talk) 15:56, 13 December 2009 (UTC)

OK, now you are doing wrong. Why do you also removed the etymological part. It addresses the topic about intersecting the asymptote. A huge amount of discussion, in the talk page is devoted to that point. It is important to be said in the article. Besides it is supported by references both the literal etymology an the use in old books. That deletion was unnecessary and decreases the value of the content of the article.  franklin  14:20, 13 December 2009 (UTC)


 * But the etymology is still there.  Sławomir Biały  (talk) 14:22, 13 December 2009 (UTC)


 * Never mind. You were moving around things.  franklin   14:22, 13 December 2009 (UTC)


 * OK, now about the quantitative information stuff? I don't see that one move but removed.  franklin   14:25, 13 December 2009 (UTC)


 * Yes. Removed.  See explanations on talk.  Sławomir Biały  (talk) 14:28, 13 December 2009 (UTC)


 * Saw it already and there is an answer to that also. Meanwhile the topic is resolved is better if the information stays in the page as it is in the very least referenced.  franklin   14:40, 13 December 2009 (UTC)


 * You have said many times that you would prefer it if editors would "fix" the article rather than discuss it on the talk page. That is what I am trying to do.   Sławomir Biały  (talk) 14:42, 13 December 2009 (UTC)
 * Definitely, do it. But I also will look for the relevant information to be preserved.  franklin   14:44, 13 December 2009 (UTC)


 * OK now you are getting unreasonable. The paragraph comes from an idea expressed in a book. Please don't remove information change it if you don't like the wording.  franklin   14:46, 13 December 2009

(UTC)
 * You see? I am happy now. It would be good to also add further down in the article more detain on how that quantitative info is used and is different from the other several notions of limiting behaviors.  franklin   15:05, 13 December 2009 (UTC)


 * Please look at what you do. I don't have strong feeling about mentioning the classification in the intro, but in your revert you put qualitative again instead of quantitative. And there is a big difference. Difference which is the main point of that sentence.  franklin   15:42, 13 December 2009 (UTC)


 * Also now the part after the etymology only talk about the use of asymptotes as tangents at infinity but not about intersecting (they are different things and both should be addressed since both lead to a lot of discussion and confusion in the talk page and in the history of the article)  franklin   15:47, 13 December 2009 (UTC)
 * solved. good. better wording needed (since looks like an isolated phrase, a forgotten thing said at the end of a conversation) but that can be fixed.  franklin   15:53, 13 December 2009 (UTC)


 * The sections on horizontal, vertical and oblique should be subsections and should have some intro. There are notation and approach not properly introduced. That was the reason for putting the classification thing there. But it was more important to preserve the pride of having that in the intro than to make the text consistently written.  franklin   15:53, 13 December 2009 (UTC)

Are asymptotes quantitative or qualitative?
It has been repeatedly insisted by a certain highly opinionated editor that asymptotes are not qualitative, but rather quantitative. This clashes with my own feeling, given that when I used to teach freshman calculus, asymptotes were always introduced in the "qualitative analysis of functions" part of the lecture. My worry here is that the target audience are mostly going to be thinking of asymptotes as part of the qualitative analysis of functions. If that is so, then we should say qualitative rather than quantitative. Sławomir Biały (talk) 16:08, 13 December 2009 (UTC)
 * Opinionated. I like that although you probably used it as a not so good adjective. Look, asymptotes definitely give qualitative info about the curve, but so do rate of growth, asymptotic expansions, limits, monotonicity. What distinguishes asymptotes and makes them the origin of the notion of asymptotic expansions is the need for precise info about the behavior of the function to infty. Thats why it is the emphasis in the quantitative part. quantitative info always give qualitative info as well.  franklin   16:14, 13 December 2009 (UTC)


 * Still, much more important than just having the phrase there, is explaining how is that. Maybe not in the intro but that was removed and should be added to the article (doesn't matter if with a different wording as the one I used). The presence of that explanation is far more important than that phrase in the intro, that can as well be removed, since right now it says that is gives both quantitative and qualitative info, which is redundant and losses the real purpose of it.  franklin   16:21, 13 December 2009 (UTC)
 * Well, as I believe I have pointed out, the section was sourced to a French-language source which nowhere uses the term "asymptote". Also, one might take the opposite point of view to that expressed in the text, namely that whereas the limit being infinite expresses quantitative information about the blowup of the function, asymptotes describe qualitative differences in this blow up.  Anyway, the content whose restoration you are now pushing was poorly expressed, confusing, and entirely unnecessary.  (I won't use the word "pride" that you are so fond of rudely accusing others of.)  Sławomir Biały  (talk) 16:28, 13 December 2009 (UTC)


 * Just a little point. a limit that is infinity is never a quantitative info. That precisely distinguishes it from rate of growth. Again, is precisely for this kind of confusion that is necessary such an explanation. I can not emphasize enough how important the purpose of a concept is way more important than any particular definition of it. That's what happened for instance with the notions of point and line. the ancients tended to waste a lot of effort trying to explain the zero dimensionality of the point and the one dimensionality of the line, while it was never important for the work they were doing afterwards of them. (I am not saying that the concept of dimension was not developed afterwards) In a similar fashion in this talk-page ( in classes also, and books ) a lot of effort is expended in arguing if to put curves or lines as asymptotes, or to put this or that, variation in the definition. If the purpose is understood then all of this are just subtle differences.  franklin   16:41, 13 December 2009 (UTC)
 * A mnemonic trick to understand the qualitative nature of a limit that is infinite. If I tell you: I have this function that tends to infinity and this other that also does. Can I compare them in any way? More info will be needed. On the other hand, if I tell you one is as fast as $$x^2$$ and the other as fast as x then the picture change (this is the change from the notion of limit to the one of rate of growth). If I tell you one is as fast as $$3x^2+1$$ and the other as $$600000000x-10$$ then the picture changes slightly more (thats the difference between rate of growth and asymptote or asymptotic expansion). Again purpose is the key so purpose should be shown to students (and sometime not only to students)  franklin   16:52, 13 December 2009 (UTC)


 * more mnemonic here  franklin   17:39, 13 December 2009 (UTC)


 * Again, if qualitative vs quantitative is confusing for you (not to be blamed, it is for many) this is just a word, again, details. What is important is to provide an explanatory paragraph pointing purpose of the concept, how this purpose distinguishes it from other concepts etc. Why to do that? Well, notice that an important reason why this article was so poorly written is that the boundaries of the notion of asymptote are not very clear, (it is just an old concept that nowadays is only useful in curve tracing in introductory courses if we talk only about lines. But not widely treated with that name if we look at other variations.). A paragraph like that will not only give information and guide the readers it will also protect the article from being changed drastically and from having disputes from those who have seen the concept written in a way or another. To do so, it would be good to explicitly, implicitly or linking to answers of questions like:
 * 1) How and why asymptote and asymptotic are linked.
 * 2) How is it different from other notions (notions that are different but are also very similar as rate of growth or maybe limit) [if other are found they are welcomed to be added]
 * 3) How is asymptote used (which is different to how is it computed) This tell about why is it computed at all.

My signature looks better down here.  franklin  17:34, 13 December 2009 (UTC)


 * Not sure how you managed to get the misconception that I am confused about something. Just because I disagree with you doesn't automatically imply that I am somehow deficient, prideful, crying, etc.  Thanks,  Sławomir Biały  (talk) 17:39, 13 December 2009 (UTC)


 * Definitely not, I am using sources to support my claim. Follow the link above. I said confused because you first called asymptote qualitative, then limits that are infinite qualitative, then asymptotes qualitative and quantitative (this one correct but rather vacuous statement). If you have clear this notions please use them to improve the article. Right know the phrase used is saying nothing, it is not playing the role that was supposed to play when added at the very beginning. But please don't call infinite quantitative, it is painful. It is like when kids try to subtract two infinites. (there it is, another mnemonic trick. you can't subtract infinites without using more info about the functions, but you surely can if you use asymptotic information about them)  franklin   17:40, 13 December 2009 (UTC)


 * As I have said repeatedly, analysis of a function's asymptotes is one of the main parts of the qualitative analysis of a function (e.g., in graph sketching). This isn't controversial stuff: look in any textbook.  But you have repeatedly insulted me here.   Sławomir Biały  (talk) 18:51, 13 December 2009 (UTC)


 * A third opinion, preferably someone with a native command of the vagaries of English, who has (hopefully) taught a course on this very subject, would be helpful. Sławomir Biały  (talk) 18:53, 13 December 2009 (UTC)


 * Please, what book? Notice that in order to sketch a graph you need also quantitative info in addition to the qualitative one. That's precisely the reason why you compute asymptotes and not only the limit at infinity. That you use the word "vagaries" is good, it show that you are getting to the limit to which you can support your point. The thing is that you are trying to come with and explanation using only the image you have in your mind about qualitative vs quantitative. The link I gave you above is taken from a book dedicated to that, made for statistics and related areas or research. In any case, qualitative, quantitative, is just a word. It doesn't matter if you don't understand, if you don't want to understand or if you are the top most authority on the subject. What is important is to answer the questions enumerated above (which is not done in the article), using or not the words quantitative or qualitative. About offending you, if telling you that you are not understanding something, giving you references and explanations that show that you are is offending, yes I am offending and I will keep doing it. Truth is more important than any one's pride. About looking in "any book". Almost no textbook qualifies asymptotes as being one or the other. I gave you one that does (the one in French). Agree that they are talking about asymptotics, but the article (as it is now), in the definition of asymptote as general curves allows the possibility of identifying the two concepts. Therefore the reference in French is applicable.  franklin   19:13, 13 December 2009 (UTC)
 * Re "which book". Try Apostol, or Stewart, or Hughes-Hallet.  Any one of the standard books from which calculus is currently taught at a freshman level in any major US university will do.  Sławomir Biały  (talk) 19:40, 13 December 2009 (UTC)
 * Re "allows the possibility of identifying the two subjects". This would be WP:OR.  We should say what sources do say, not what we think they could or ought to say.   Sławomir Biały  (talk) 19:43, 13 December 2009 (UTC)


 * The thing is that you simply don't know what is the difference between qualitative information and quantitative information. Or don't want to, since I already gave you links to entire book on the subject. No problem, if you enjoy staying in your ignorance is your decision. In any case, it is not my interest to call asymptotes either way, since, as I said, that is not what is important (apart from calling them and limits that are infinite qualitative which is wrong). What I do think is important is to address explicitly the question enumerated above. Oh! one last thing. That asymptotes are used for sketching graphs and that sketch are rough approximations doesn't say that they use only qualitative info. OK, enough throwing in your face the things you don't know. I like the section with the projective stuff. That can lead to put, what someone has been pushing to put and that is useful, about computing asymptotes for curves given implicitly.  franklin   19:16, 15 December 2009 (UTC)

To me, it's not really necessary to specify either way and since it seems to be a point of contention perhaps it's best not to use either term.--RDBury (talk) 22:01, 13 December 2009 (UTC)

One of the most important steps
Stewart's Calculus gives a table of the "most important steps" for sketching the graph of a function. Finding the asymptotes is one of them. Is there a serious contention that this is not one of the most important steps? Sławomir Biały (talk) 19:48, 13 December 2009 (UTC)
 * Look, if a term deserves a page in Wikipedia it is important. If a term, among other in a list, deserves to be called one of the most important that need a reference. I don't know the name for that guideline of Wikipedia. Giving Apostol as a reference was not good as he doesn't give such a claim. It is even more preferable to have a good reference for that since, in some courses (very basic though) the computation of asymptotes is skipped. Notice that for having a rough idea of the graph of a function they are not necessary. (by the way this is a difference from qualitative notions such as monotonicity). Now you are giving Stewart. Good! If it is good. I will check and if it says it explicitely I will leave it there. I just want this article to be perfect, or close to.  franklin   20:04, 13 December 2009 (UTC)


 * do you have the page? Uff the Stewart is huge!!  franklin   20:06, 13 December 2009 (UTC)


 * Calculus, 4th ed., p. 264. I don't suggest actually using this as a reference for the text because the precise contents are in constant flux, so it is difficult to verify precisely.  I also don't think it is a particularly controversial point whether it is indeed "one of the most important" aspects: the fact that Apostol does it in all of the examples is fairly convincing to me.   Sławomir Biały  (talk) 20:13, 13 December 2009 (UTC)


 * OK, now someone pointed me to the Wikipedia policy. The thing is that you what to argue and do some things without knowing and don't like to listen. Read here please. I read Stewart. In fact it lists the computation of asymptotes as one of the steps to sketch a graph but it doesn't say explicitly that is "one of the most important". I as I told you, I really doubt that any book is going to have such a claim. In courses in which, for example, limits are not studied (as in some junior high school or courses for people very far from using math) the sketch of graphs only include things like translating and dilating graphs of known functions, with maybe determining monotonicity by very elementary methods. It is hard to talk about asymptotes without limits. Therefore, please, don't confuse being important with the claim "one the most important". After all, either of the two claims is not giving any kind of important information to the reader. It is the difference between saying "Napoleon is one of the most important characters in history" and "Napoleon conquered most of Europe, was a charismatic military leader,....". The last one gives facts, important information, the former... (apart from being discouraged by Wikipedia) is not saying anything.  franklin   14:44, 15 December 2009 (UTC)

Progress
It looks like there has been a lot of progress here. I still think there there is a bit too much emphasis on asymptotes of functions (as opposed to curves in general) which could be covered just as well in the Asymptotic analysis article, but that may just be me and the highest priority was to get references for everything.--RDBury (talk) 11:14, 15 December 2009 (UTC)
 * It is getting pretty. Can we nominate it for Featured Article when done? I have to read what are the criteria for that to try to achieve it. It would be so nice that an article goes from such a deplorable state to featured in so short time.  franklin   14:56, 15 December 2009 (UTC)
 * Was reading the guide lines but the article will be needed to let rest to satisfy the stability criterion, 1e. after checking the rest of the criteria and before nominating.  franklin   20:37, 15 December 2009 (UTC)
 * I would not get your hopes up as I doubt it's ever going to make an FAC: even when it's done it's going to be mostly mathematical formulae and technical discussion, of little interest to general readers. The high density of maths would make it difficult to provide reference and fill it with the sort of sparkling prose that seems to be expected. Look at WP:FFA; none of them look anything like this one.--JohnBlackburne (talk) 21:10, 15 December 2009 (UTC)
 * I see. Probably very true. Pity. Also, compared to the eigenvalue one, this is a piece of garbage right now. After all, asymptote is just an old not-very-useful concept that evolved into others of greater significance.  franklin   21:27, 15 December 2009 (UTC)
 * I'll be more than happy if we can get it to B or B+ class. Right now I'd say it's gone from barely Start to about C, but math doesn't do C's so we can't change the rating yet. I've been reading dozens of different texts online to get an idea of the different definitions being used, and it seems like everyone has their own variation. So it's going to take a while to get it all to make sense.--RDBury (talk) 08:10, 17 December 2009 (UTC)
 * Is that an official classification going on in Wikipedia? Do you have a link to the requirements? I would like to read about to see what can be done if it can be done. About the concept it self, it is not such a big problem that there are many variations. The article can simply talk about them. Only two things will be needed: 1) try to talk about all main variations so that no one feels excluded and tempted to start changing drastically the article. 2) try to put them with the appropriate hierarchical emphasis according to popularity as it is required by Wikipedia. I was making so much emphasis in showing the purpose of the concept and the relations between the different presentations because of this. When the purpose is stated all variation in the concept become just details and people using one variation will understand (and respect) the variations of the others. I remember when in high school that they never agreed if to put zero as a natural number or not. Fortunately there was no doubt about having induction and that shows where the main purpose was.  franklin   12:02, 17 December 2009 (UTC)

Simple example
I added an example and some additional explanatory material to explain the concept to non-mathematicians. The example does require some familiarity with the Cartesian coordinate system, setting the level at pre-calculus or high school. WP:MOSMATH recommends an informal introduction of this type and there was none. The text was taken, at least in outline, from the "Asymptote" article in the Penny Cyclopaedia, a popular 19th century encyclopedia, but it seemed as good an introduction as any with a bit of updating and it's freely available on Google. I translated the water example given there to one using a curve defined in Cartesian coordinates because, as given, the original would be hard for modern readers to follow and it would require additional diagrams.--RDBury (talk) 12:19, 18 December 2009 (UTC)

Asymptotic curve
Most of what I see is asymptotic lines and no asymptotic curves. I am sure you know what I mean. Let f(x) = 1/x, g(x) = x^2 be the asymptotic curves, and let h(x) be the function. So, can someone place drawing of one on this page? --John W. Nicholson (talk) 05:18, 17 December 2013 (UTC)
 * Like the drawing in the section on curvilinear asymptotes? Sławomir Biały  (talk) 12:03, 17 December 2013 (UTC)
 * Almost, I am thinking of one which does not have any linear asymptotes, but curves. I realize my statement above does not do that with 1/x. Also, there is nothing about primes with asymptotes, maybe add something with it. — Preceding unsigned comment added by User: (talk • contribs)
 * What do you mean "primes with asymptotes"? I'm not convinced that a different illustration is needed for a curvilinear asymptote. This shows that you can have lines as well as more general curves as asymptotes: they are not mutually exclusive.  Sławomir Biały  (talk) 16:18, 17 December 2013 (UTC)

Relation to "line of best fit"?
Intuitively, it would seem to me that asymptotes and lines of best fit are related concepts, but with lines of best fit typically describing a line that is minimal mean distance from a set of points in a field, while an asymptote is a line of infinitesimally small distance from another line as both lines tend to infinity; essentially, both concepts describe the line that is minimally distant to another object at some place.

Are there cases in which the line of best fit could be described as "asymptotic" or vice versa, or where the concepts formally overlap? Or is there something else key that I'm not grasping? — Sasuke Sarutobi (talk) 13:04, 18 February 2016 (UTC)

Assessment comment
Substituted at 01:47, 5 May 2016 (UTC)

Image quality
Images are of highly variable legibility. Some are presented well in clearly-distinguishable, highly-contrasting colours. Others are very difficult to read, being presented in various scarcely distinguishable shades of grey upon grey (as promoted by modern graphics designers, apparently). —DIV (120.17.226.223 (talk) 00:14, 1 April 2017 (UTC))