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This question is coming from the analytic geometry viewpoint. Please ignore the viewpoint of algebraic geometry here, unless that viewpoint is somehow able to handle non-algebraic curves like $x \mapsto e^x$ etc. which I don't think it is.

According to my Year 12 mathematics textbook, an asymptote is:

A line that approaches a curve but does not touch it.

With the benefit of the modern viewpoint, this definition just feels very weird and 18th Century, for the following reason: to "approach a curve" is to be somehow "close to it"; but, according to the above definition, if we get so close to the curve that we actually intersect it, then we're "too close" and its no longer an asymptote. Presumably, this isn't desirable, and so the hope is that modern authors no longer include this condition in their definition of "asymptote."

A moment of Googling brings up the relevant Wikipedia page:

In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors.

Okay, this is definitely making a lot more sense. But, I still think a few things are screwy. Firstly, there are different definitions of curves lying about, and they aren't completely consistent. So lets just get rid of any mention of curves at all. We obtain:

Let $A$ denote a subset of $\mathbb{R}^2$. Then an asymptote of $A$ is a line such that the distance between $A$ and the line approaches zero as they tend to infinity.

Alright. Its still imprecise. Notice we haven't told the reader what exactly is approaching infinity. There's at least two ways of resolving this ambiguity, and they result in two fundamentally different notions of "asymptote."

Asymptote, Undirected Version. Let $A$ denote a subset of $\mathbb{R}^2$. Then an asymptote of $A$ is a line $L \subseteq \mathbb{R}^2$ such that for all $\varepsilon \in \mathbb{R}_{>0}$, there exists $r \in \mathbb{R}_{>0}$ such that for all $l \in L$, if $d(l,0)>r$, then the distance $d(l,A)$ is less than $\varepsilon$.

For example, the function $x \mapsto e^{-x^2}$ has precisely one asymptote according this definition. And the function $x \mapsto e^x$ has no asymptotes at all.

There's also another possible way of resolving the ambiguity. First, we need an auxiliary definition.

Directed Line. A directed line in $\mathbb{R}^2$ is a function $c : [0,\infty) \rightarrow \mathbb{R}^2$ such that:

  • $c$ is injective
  • $c$ preserves convex combinations: that is, for all $a,b \in [0,\infty)$ satisfying $a+b=1,$ we have $c(ap+bq) = ac(p)+bc(q).$

Two directed lines are said to be equal iff they intersect at more than one point.

We can then define asymptote like so:

Asymptote, Directed Version. Let $A$ denote a subset of $\mathbb{R}^2$. Then an asymptote of $A$ is a directed line $c : [0,\infty) \rightarrow \mathbb{R}^2$ such that for all $\varepsilon \in \mathbb{R}_{>0}$, there exists $t \in \mathbb{R}_{\geq 0}$ such that for all $t' \geq t$, the distance $d(c(t'),A)$ is less than $\varepsilon$.

For example, the previously-mentioned function $f : \mathbb{R} \rightarrow \mathbb{R}$ given by $f(x)=e^{-x^2}$ has precisely two asymptotes according this definition. And the function $x \mapsto e^x$ has precisely one asymptote.

Now clearly, the undirected definition is easier to state. But I feel the the directed one is more useful. Anyway, my question is simple:

Question. How are asymptotes actually defined in rigorous mathematics?

I am not suggesting that there can be only one definition. If multiple definitions are useful, then I'd like to know about all of them. Thanks.

goblin GONE
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    I've personally never seen a definition of asymptote for anything else than the graph of a function $\mathbb{R} \to \mathbb{R}$. – Siminore Aug 25 '15 at 09:30
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    "Asymptote, Undirected Version." looks a bit weird. Where do you use $r$? – Colm Bhandal Aug 25 '15 at 09:38
  • @ColmBhandal, thanks! Both definitions were broken hehe. I think they're fixed now. – goblin GONE Aug 25 '15 at 09:50
  • @Siminore, I'm not quite sure what you mean, but note that $x \mapsto 1/x$ does not define a function $\mathbb{R} \rightarrow \mathbb{R}$ under the usual definitions, because functions are by definition total. Even putting this aside though, I'm not quite sure what you mean. Most people would probably say that $x \mapsto 1/x$ has both ${(x,y) \mid x=0}$ and ${(x,y) \mid y=0}$ as asymptotes. Are you saying that, under your preferred defintions, the first of these isn't a "real" asymptote? – goblin GONE Aug 25 '15 at 09:54
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    My notation intended to denote "real functions of a real variable", sorry for the confusion. In my experience $x \mapsto 1/x$ has exactly two asymptotes: $x=0$ and $y=0$. I have never seen a definition of asymptote for an arbitrary subset of $\mathbb{R}^2$. Your directed definition recalls of the definition of oblique asymptote, anyway. – Siminore Aug 25 '15 at 09:58
  • @Siminore, okay, I see what you mean. Honestly I think that subsets of $\mathbb{R}^2$ are the right viewpoint here, for the following reason: if you rotate a function about the origin, it needn't remain a function; but, obviously, its asymptotes should rotate with it. – goblin GONE Aug 25 '15 at 10:01
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    But, without any restriction on the subsets, every line is an asymptote of $\mathbb{R}^2$, which doesn't seem very useful. It may be worth paying that price if characterising "sufficiently curve-like" subsets is too cumbersome. – Daniel Fischer Aug 25 '15 at 10:03
  • @DanielFischer, this just comes back to the need to find out how asymptotes are actually used in rigorous mathematics. It may be that having "too many" asymptotes isn't a problem, or it may be that it is a problem. Imo we cannot expect ourselves to figure out whether or not this is a problem with only a few minutes (or hours) thought. – goblin GONE Aug 25 '15 at 10:05
  • @goblin Let me ask a question. Consider the set ${(x,x \sin x) \mid x \in \mathbb{R}}$. Would you like to give $y=x$ and $y=-x$ any role? – Siminore Aug 25 '15 at 14:35
  • @Siminore, my intuition says they aren't really asymptotes, they're just upper and lower bounds. This is consistent with the "directed" definition. On the other hand, my intuition also says that $x \mapsto x \sin (x^2)$ has no asymptotes. However, this is inconsistent with the "directed" definition – goblin GONE Aug 25 '15 at 14:42
  • I like your "curve-free" definition, but I'm not sure whether asymptotes are commonly used (or even used at all) in modern mathematical research. – Kyle Strand Aug 25 '15 at 15:58
  • In your directed line definition, should the f's be c's? – snulty Aug 25 '15 at 18:06
  • @snulty, thanks, fixed. For the record, I'm profoundly dissatisfied with my directed line definition (even the fixed one!). Its just something quick and dirty I threw together for the question. Presumably, there a better, sleeker definitions of "directed line" available. – goblin GONE Aug 25 '15 at 18:09
  • @goblin if you're trying to get half lines why not say, if it starts at a point $p$ in the direction of $q$, with $p,q \in \Bbb R^2$, it's the function $c:[0,\infty)\rightarrow \Bbb{R}^2$, $c(t)=p+t\cdot q$, which uses the vector space structure of $\Bbb R^2$. Or is it more general than that? – snulty Aug 25 '15 at 18:17
  • @snulty, yeah, it's the same thing. The dissatisfaction is mainly in the phrase: "Two directed lines are said to be equal iff they intersect at more than one point." These kinds of definitions make mathematics more painful than it needs to be. – goblin GONE Aug 25 '15 at 18:19
  • @goblin You know if you modify the answer I presented you could change it from curves as functions to arbitrary unbounded sets. Just change from $A\subset \Bbb R$ to $A\subset \Bbb R^2$, and drop the mention of $c$ entirely? – snulty Aug 25 '15 at 18:45
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    Notice that, under your tentative definitions, the function $\sin(1/x)/x$ has (if I'm not mistaken) a vertical asymptote at $x=0$, which does not agree with common definitions (vertical asymptote $\iff$ $\lim f= \pm \infty$). I'm guessing that it's almost desperate to look for rigourous (and widely accepted) definitions of asymptotes because they hardly belong to "rigorous mathematics". – leonbloy Jun 15 '19 at 13:04
  • @leonbloy, woah! That's now my favorite function :) – goblin GONE Jun 15 '19 at 14:02
  • @goblin Inspired by this https://math.stackexchange.com/questions/1863341/if-fx-has-a-vertical-asymptote-does-fx-have-one-too/1863412#1863412 – leonbloy Jun 15 '19 at 14:05

2 Answers2

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How does the following work. Let $A\subseteq \Bbb R$ be an interval and $c: A \rightarrow \Bbb{R}^2$ continuous such that $c(A)$ is unbounded in the sense of the usual metric $d$ on $\Bbb R^2$. We can certainly find a sequence $a_n$ in $A$ such that for each $r>0,\,\exists N \in \Bbb N$ such that for $n>N$ we have $c(a_n) \not\in B_{r}((0,0))$. Now let $L\subset \Bbb R^2$ be a line, we say $L$ is an asymptote to $c$ if $\lim\limits_{n\rightarrow\infty} \mathrm{inf}\{d(c(a_n),l)\mid l\in L\}=0$, for a sequence $a_n$ as defined above.

Note you then add restrictions on how many times the curve is allowed to intersect a Line. Otherwise a Line is an asymptote to itself, and in fact any line is an asymptote to a spiral centered at the origin, eg Archimedean spiral.

snulty
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  • Obviously, this is more sensible and better behaved than either of the two definitions that I give. However, I'm looking for accepted definitions here, not innovative or clever guesses at what the definition should be based on a few minutes or hours of thought. I'm not saying that I want this deleted (I most certainly don't), but I am saying that this doesn't answer the question. – goblin GONE Aug 27 '15 at 12:42
  • I understand your point! I agree it doesn't really answer the question asked. However, a scan for asymptotes gives the following arrive paper about generalised asymptotes, where you allow the asymptote to be a curve rather than a line. @goblin - http://arxiv.org/pdf/1307.6153.pdf – snulty Aug 27 '15 at 12:56
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I think this may be more what you're interested in here. The author is Herbert Busemann, and the paper is on Local Metric Geometry, and he mentions asymptotes in metric spaces, where the distance is not necessarily symmetric, and according to the reference here on project Euclid, by Nasu, he says the concept of asymptotes was introduced. Check his references 2 and 3. That's one of them but I couldn't find the acta math one

snulty
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