# Teaching limits and asymptotics at the same time

Having never been a mathematics educator, my question could be stupid and, if this is the case, please delete it.

When I was young, from the very beginning of limits, we were teached that there are two things to look at simultaneously : what is the limit and how it is approached. This means that, even when quite young, the concept of asymptotics was in our minds and, thinking back, I have the personal feeling that this was a great idea.

Visiting almost daily MSE, I have the feeling that this is not the case almost anywhere.

So my question : am I totally out of date ? Should I be wrong if I tried to teach that way ?

• See my answer at matheducators.stackexchange.com/a/10077/1550, which contains an explanation of this idea in a way that can be taught to high-school students, but sadly everyone who has read it either don't appreciate it or don't even understand it... – user21820 Nov 26 '15 at 8:31
• Thanks for answering. Your answer is very interesting. In fact, not being an educator, what I find funny is that there is no conceptual problem when trying to teach asymptotics and/or Taylor expansions. And, from there, everything becomes so simple when we have to deal with limits ! – Claude Leibovici Nov 26 '15 at 8:40
• Exactly. Perhaps as a non-educator you have a better understanding of what students (who are also non-educators) face in understanding limits! =) – user21820 Nov 26 '15 at 8:41
• And I should add that asymptotic analysis is so very natural because it follows straight from generalizing linear approximations (which occur for all differentiable curves). – user21820 Nov 26 '15 at 8:43
• I don't know how young pupils/students percieve mathematics outside France. Here, they consider that it is a punishment to suffer and that it is of no use. When I show that mathematics can lead to beauty and even fun, most of the time, I suppose they look at me just as if I was coming from Mars. – Claude Leibovici Nov 26 '15 at 8:47

## 2 Answers

If I understand you, I do often think this way when solving a problem. For example, see this answer of mine at Mathematics Stack Exchange.

However, limits involving infinity (as the independent or dependent variable) seem to be losing importance in textbooks. For example, I teach from Calculus: Graphical, Numerical, Algebraic by Ross L. Finney et al., and that book sticks "Limits Involving Infinity" into one section and largely ignores the concept otherwise. This approach has some advantages, such as not needing to distinguish between a limit of plus-or-minus infinity and an undefined limit. But you see it has some disadvantages as well.

• I know you for quite a while and, from answers at MSE, I think that we share many points of view on problems. Thanks for taking the time of answering. – Claude Leibovici Jul 1 '15 at 10:36
• I feel the idea of focusing at first only on finite limits (at a finite value of $x$) is justified on pedagogical grounds. The reason is that students are bored with limits in the initial stages, and invariably all we do in studying limits initially is to rewrite expressions in a different form that makes the limit obvious by substitution. It is difficult for students to understand what the substance is in this formalism. The idea of this approach is to restrict the study of limits to the bare minimum needed to define and compute derivatives, returning to subtler questions... – Keith Jul 2 '15 at 1:00
• (cont'd) including infinite limits and limits at infinity, in the context of curve sketching. – Keith Jul 2 '15 at 1:01

I'd recommend not being intimidated by textbook trends! "In real life" (as opposed to textbook-life, for sure, and often opposed to required-curriculum "school-math"), _of_course_ the two things you mention, "the limit", and "how it is approached", matter a great deal. Do not be intimidated by silly books (written by non-mathematicians, almost entirely) to think otherwise. That is, _of_course_ we care about the asymptotics as you mention. True, "sadly" for immature students, those details are more complicated than the more-typical (in the U.S.) symbol-pushing, and it would be hard to convince the kids to pay attention at all if "it's not on the final", ... which it cannot be, mostly, because it is not the general standard.

But, yes, there is an underlying reality of human understanding of these things, in which asymptotics play a large role...