# 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So my question : am I totally out of date ?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

• 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