# Educational resources commonly address slant asymptotes. Why not general polynomial asymptotes?

*Answer: It's not very useful (not in practice, not in theory).*

*Cross-posted from here.*

## The Question

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.”)

It struck 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).

*Why are polynomial asymptotes not normally covered by resources that teach slant asymptotes?*

## 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.

## Accepted Hypothesis: It's Not Very Useful

*It’s not very useful in practice.* 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.

*It’s not very useful in theory.* 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.