Talk:Asymptote

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[edit] 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.

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]

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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 α when there is a continuous mapping γ of [0,1) into the complex plane such that γ(t)→∞ and f(γ(t))→α as t→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 ε, 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 ε.

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 ε, 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 ε.
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=nπ, 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)


[edit] 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)

[edit] 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)