5
$\begingroup$

Back in 2018, I wrote a post about asymptotes of rational functions in which I addressed not only horizontal and slant/oblique asymptotes, but also the general case of "polynomial asymptotes." Fast-forward over 5 years later, the post gets plenty of traffic from the query "polynomial asymptote" because apparently polynomial asymptotes are not covered anywhere else! (Below is a snippet of the post that describes what I mean by "polynomial asymptotes.")

enter image description here

It strikes me as really weird that polynomial asymptotes are not covered elsewhere. They are a natural generalization of slant asymptotes, they help develop intuition about why graphs of rational functions look like they do, and they don't require any more advanced computation techniques –- it's still just polynomial long division (or even synthetic division if your divisor is linear).

My questions:

1. Why are polynomial asymptotes not normally covered along by resources that teach slant asymptotes?

2. I can't find the phrase "polynomial asymptote" elsewhere on the web. I tried searching images too and don't see any other images of graphs illustrating this concept. Is the phrase "polynomial asymptote" truly nonexistent outside of this post? If so, why? If not, what other resource(s) mention it?

My intention with these questions is less about critiquing curriculum and more about just trying to make sense of a phenomenon that strikes me as really weird.


Falsified Hypotheses

Below are some explanations that sounded initially sounded promising, but upon deeper analysis, turned out not to be true.

  • Maybe it's because people just refer to "slant asymptotes" in the most general sense, not just lines but also polynomials of degree greater than 1. False. Resources that talk about slant asymptotes generally say that "a slant asymptote occurs when the degree of the numerator is 1 more than the degree of the denominator." So I don't think that's the case.

  • Maybe it's because polynomial asymptotes are just one special case in general asymptotic analysis. False. In the context of rational functions, polynomial asymptotes fully describe the general case -- they're not just a special case. And by this reasoning, it should also be uncommon to address slant/oblique asymptotes, which are actually special cases.

Non-Falsified but Non-Definitive Hypotheses

Below are explanations that sound promising and have not been falsified, but have not accumulated enough evidence to be considered definitively "the answer."

  • It's less useful in practice than it is in theory. In practice, sketching a general polynomial is not that much easier than sketching a general rational function. And often we only care about behavior for large x, which we can easily determine by just dividing the highest-degree term of the numerator by the highest-degree term of the denominator. True, but not definitive because educational resources often cover other things that are less useful in practice than they are in theory.

Update: The accepted answer makes the above hypothesis definitive by pointing out that there's actually not much theory that builds on polynomial asymptotes, whereas there is more theory that builds on related topics of little-o and big-O notation. So if you start with polynomial asymptotes and follow that line of thought deeper, you end up at little-o and big-O notation, for which polynomial asymptotes are not actually a prerequisite.

$\endgroup$
17
  • 1
    $\begingroup$ As a point of reference, I wasn't taught, nor have taught, slant/oblique asymptotes. If I had been taught them, it would have been about 45 years ago. Horizontal asymptotes have quite few ties to applications; for instance, equilibria in DEs. The nearest thing I know to polynomial asymptotes are limit cycles and transients in oscillatory systems. -- It disappointed me to figure out later that functions that are asymptotic to each other are not "asymptote-ic" to each other. (For instance, if $y=ax+b$ is an asymptote of $y=f(x)$ then $\lim f(x)/(ax+c)=1$ at infinity for any $c$.) $\endgroup$
    – user1815
    Commented Oct 31, 2023 at 15:05
  • 1
    $\begingroup$ I can't find the phrase "polynomial asymptote" elsewhere on the web. --- See my first comment to How do we know that a rational function has at most 1 slant asymptote? (and my comment to the answer), this 22 May 2003 sci.math post and these google hits. I probably used the term often in the ap-calculus discussion group when it was hosted by Math Forum, but those posts are no longer available. $\endgroup$ Commented Oct 31, 2023 at 15:55
  • 2
    $\begingroup$ I don't know why it's hardly taught. It provides reinforcement long division practice as well as showing an application that students can appreciate, and you often get neat looking graphs when graphing by calculator or computer algebra. I've seen this sometimes covered in college algebra and precalculus texts, but with other terms (I listed some of these terms in another comment at the SE link I gave). I don't think I've ever seen the term "polynomial asymptote" used before I made up the name, but surely others must have used the term because it seems such an obvious term. $\endgroup$ Commented Oct 31, 2023 at 16:29
  • 1
    $\begingroup$ It seems the only term I mentioned is "nonlinear asymptote". I remember seeing a rational function with a cubic asymptote in a college algebra/precalculus text I taught from around 1993-1996, but I can't find it now for some reason. In fact, the graph I'm thinking of (I remember how it looked) isn't in any of the books I looked through just now on my bookshelves, but I did find two that bring up the notion without naming it: Burzynski/Ellis/Lodi, "Precalculus with Trigonometry" (1995; worked example on pp. 267-270) AND Cohen, "College Algebra" (4th ed. 1996, exercise 9 on pp. 343-344). $\endgroup$ Commented Oct 31, 2023 at 16:52
  • 2
    $\begingroup$ I think it's the nature of the subject, so many students struggling to stay above water. I don't even try slant asymptotes with the College Algebra crowd. If I can get them to deal correctly with horizontal asymptotes then I count it a blessing. Of course, we ought to teach polynomial asymptotes, just like we ought to teach many many things. $\endgroup$ Commented Nov 2, 2023 at 2:53

4 Answers 4

3
$\begingroup$

I think it is just that the related concepts encompassed by little-$o$ and big-$O$ notation are more important than "polynomial asymptotes" and do find many applications. We do teach Landau notation in analysis, computer science, etc.

When we talk about asymptotics, we typically reference just the dominant term. For examples, if a function is $O(n^2+n)$ then it is $O(n^2)$.

I remember when I first taught college algebra after my PhD. I don't think I had ever encountered slant asymptotes in graduate studies, and I embarrassed myself in front of that class many years ago by saying that if $y=x+1$ is a slant asymptote then saying that $y=x$ is also a slant asymptote is equally correct. But of course I was confusing the statement that $O(x+1)$ and $O(x)$ are the same with the more geometrical notion of a unique slant asymptotes.

It also occurs to me that in algebra, meaning the world of polynomials and rational functions, the notion of a polynomial asymptote is not very helpful since it says nothing about polynomials. It does have some teeth with rational functions, but then with rational functions we typically focus only on "end behavior." For example, we look at $y=\frac{x^2+1}{x+1}$ and see that it behaves like $y=x$ at infinity, and don't bother with the refinement that it behaves more like $x-1$.

$\endgroup$
1
  • 1
    $\begingroup$ Thanks, this is convincing to me. Before reading this answer, the most promising hypothesis was that "it's less useful in practice than it is in theory," and this answer points out that there's actually not much theory that builds on it, whereas there is more theory that builds on related topics of little-o and big-O notation. So if you start with polynomial asymptotes and follow that line of thought deeper, you end up at little-o and big-O notation, for which polynomial asymptotes are not actually a prerequisite. $\endgroup$ Commented Nov 1, 2023 at 22:47
5
$\begingroup$

This is kind of a joke answer, but in my favorite math story ever we have the following exchange:

Eric pondered a moment. "But... but if it's that simple, why don't my textbooks talk about it?"

Grinning evilly, the Wiz replied: "They certainly drop the necessary clues. As for why they don't emphasize this stuff, well, this is just one of those tricks we Wizards use to distinguish the people who think for themselves from those who fall for any plausible line of claptrap. In fact, every time Halloween falls on a full moon - like this year - we get together and agree on what facts like this we will keep secret, precisely to see who rediscovers it for themselves. This year we...." At this point the absent-minded Wiz caught himself. "Whoops! This is too secret for you! Anyway, you just failed one of these tests."

I certainly enjoyed discovering polynomial asymptotes as a high school student, and it looks like several people in the comments to the OP did as well.

Maybe it has not made its way into the standard curriculum because it is not essential to any further coursework, but is "low hanging" enough that talented students get to discover it for themselves. The marginal utility of introducing it to the curriculum is negative since:

  1. Students who don't care about math will not be excited by one more thing to memorize, so less utils for them.
  2. Students who do care about math will get less utils out of it when presented as "known math" vs. if they discover it for themselves.
$\endgroup$
0
$\begingroup$
  1. For issue 1, I suspect that it is just an issue of time benefit, like we could think of many enrichments. I don't think it is an issue of nobody thought of it. Probably also contributing are the issues that the solutions are not a simple generalization of the line assymptote finding.

    It may not be exactly be in analytic geometry class, but I do think students look at this behavior in calculus and natural sciences. So, the limit of x plus e to the x is e to the x for instance, as x grows large. Also in ODEs there is some looking at solutions that we can't express analytically but that we can bound or show approaching curves.

    Possibly another issue is drawing the parabola or what have you that is the limit curve. The whole point of curve sketching is to abstract a few features, intercepts snd assymptotes snd symmetry snd the like, that allow sketching a curve without just plotting z gazillion points. But if your bounding limit is a parabola, to use it well, you'd need to have a good sketch of the parabola itself. Very different than physically drawing a dashed line on graph paper with a straight edge.

  2. I'm not so sure about the term polynomial assymptote being appropriate. If you look at the wiki article on assymptores they talk about curve assymptotes. And you think of hyperbolic or exponentials or trig functions or the like that we might approach with given expressions. Anyhow, I would loosen up the search terms and see what you get.

P.s. Just sharing a few thoughts, not a definitive answer, so mods feel free to commentize, although really the question is multifaceted. And honestly I chafe at the whole idea of lock and key answers and a semantic web. Think forum style discussion is a feature not a bug.

$\endgroup$
0
$\begingroup$

A guess: Could it be a matter of time constraints? Thoroughly covering all potential outcomes and their implications is a time-intensive endeavor.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.