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I understand that the asymptote to a curve is a straight line such that the distance between the curve and the line tends to zero as they tend to infinity. However many books also say that an asymptote is a straight line which meets the curve at two coincident points at infinity. My doubts are:

  1. I can't understand the second definition. What does meeting the curve at infinity mean? What does two coincident points at infinity mean?

  2. How are these two definitions equivalent?

I will be grateful if someone clarifies my doubts.


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In infinity, the distance between the line and the asymptote is 0, as you state. When the distance is 0, two lines touch/intersect. – Hidde Feb 20 '12 at 8:35
Think of projective curves. – Andy Feb 20 '12 at 8:37
Check: – NoChance Feb 20 '12 at 8:39
I find it hard to believe that any book would give that second definition. Please give a specific source, and copy out the exact wording, and then we'll talk. – Gerry Myerson Feb 20 '12 at 11:32
See the first line of the article here: I can't access the full article so I can't unfortunately read what the author has to say about 2 points. Whaat I want is a formal understanding of "two points". Is it that for the curve y=f(x) and the line y=mx+b; 1/(f(x)-mx-b) has a double root at zero or something like that? – Shahab Feb 20 '12 at 12:23
up vote 1 down vote accepted

I think what confuses is the fact that the definition in the book is just an informal way of saying what you said (modulo the line meeting the curve in 2 points - I'll get to that immediatly): In my expreience a lot of calculus books are deliberately imprecise so as not to frighten the student with technical definition and thus give first a definition which is supposed to appeal to the intuition. We could state the definition of an asymptote at different levels which vary in rigor and formalism. Thus we could have:

Less precise (at an essential point) and informal - your definition: "The asymptote to a curve is a straight line such that the distance between the curve and the line tends to zero as they tend to infinity" (Lack of precision concerning whether you meant only $+\infty$ or not and if not, if you meant "they tend to $+\infty$ or $-\infty$" or they tend to $ +\infty$ and $ -\infty$")

Less precise (at a not so essential point) and informal - The books definition: "An asymptote is a straight line which meets the curve at two coincident points at infinity" (Lack of precision concerning how to rigorously translate "meets the curve" into a mathematical statement)

More precise and semi-formal - Wikipedia's definition. Note that to be formal, one has to distinguish many cases in which the graph of the function could behave. Also note that in that definition arbitrary curves are excluded, since these can be rather monstruous and to ask for an asymptote for such a thing wouldn't make sense; see for example this thing ].

Note further that 1) the more formal you get, the more context you have to specify (which in the other cases is implicitly assumed. Example: You were only talking about curves of graphs of functions on arbitrary curves in $\mathbb{R}^2$) 2) these 3 levels aren't by far the only ones; one could insert many levels of rigor/formalism between these three and there would also by room. 3) definitions aren't carved in stone; different authors use slightly different definitions (which mostly vary in technicalities; the underlying intuition is almost always the same).

To address the last problem (of the first question) concerning the line meeting the curve in 2 points: This is a special case of the note 3) from above: Some authors state their one definition for various reasons. My guess, as to why the author required that the line should meet the curve in 2 points at infinity, is, again, that he wants to remain intuitive - and this definition of an asymptote just does that. The graph of the other answer illustrates this: the line given by all the point in $\mathbb{R}^2$ that satisfy the equation $y=0$ is, in his definition, an asymptote, which would also be clear by just looking at the graph (which accounts for the intuitiveness of the definition). Usually asymptotes are defined like in your definition, where one is only concerned if the line is "close enough" to the function graph either at $t \rightarrow +\infty$ (I have namend the variable of your function $t$) or $t \rightarrow -\infty$ (instead of $t \rightarrow +\infty$ and $t \rightarrow -\infty$ like in the books version), but then you would have examples of functions and lines, where the line is an asymptote to the function, but the resulting picture isn't so nice anymore, since the line would approximate" the function nicely only for, say, $t\rightarrow +\infty$.

After all this it, you should be now aware, that your definition is a little bit to vague , since you haven't specified if you meant with "infinity" only the positive infinity. If you did, the answer is no.

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  1. If the distance between the curve and the asymptote tends to infinity, where will they meet? Consider the following: $x\times y=1$. The graph is a rectangular hyperbola.

As you can see, the difference between the curve and the X/Y axis goes on reducing but it will never be 0. Or in other words, it will be 0 only at infinity.

R. Hyperbola

To make things clear : Take an example: You have 10 bucks, Every hour I will take half of them away from you. You start with 10. After an hour, you will have 5, Then 2.5, Then 1.25 and so on. When will you have 0 (not approx 0, "actual" 0) ?

The answer is at time $t = \infty$.

Also, what is happening to the difference between how much money you have and 0 ? They go on reducing and "try" to become 0 (but never will).

Hope this helped.

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You ask, in the comments, for a formal understanding of "two points". I don't think this is possible, since I don't think "two points" makes sense. The line $y=0$ is an asymptote to the curve $y=e^x$; I can see using the language of projective geometry to say that the line meets the curve at a point at infinity, but I cannot see any interpretation under which the line meets the curve at two points at infinity.

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On the wikipedia article on asympototes it is stated: "A plane curve of degree n intersects its asymptote at most at n−2 other points, by Bézout's theorem, as the intersection at infinity is of multiplicity at least two." Do you think this has something to do with this? – Shahab Feb 21 '12 at 5:57
Maybe so. Of course, when you talk about "plane curve of degree $n$" you are talking about polynomial equations, so this does not apply to the $e^x$ example. Also, note in your quote the phrase at least. That means we might be talking about two "coincident" points, but we might be talking about more than two. – Gerry Myerson Feb 21 '12 at 10:22

After reading all the answers and searching for a bit on the web I think I have found on the correct solution. I am grateful to everyone who responded. I don't think both definitions are mathematically formal. The first conveys the idea that the asymptote is a line which kisses the curve at infinity or in other words is a tangent to the curve at infinity.

The second "definition" essentially says that the tangent to the curve at infinity is of multiplicity two (The definition is silent on whether it may be more). I found this result (Proposition 6.0.3) on the internet which says that with a proper understanding of "multiplicity" at an intersection point the tangent and the curve always have multiplicity at least two. So the second statement conveys the same idea with the addition of multiplicity at the intersection.

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