How should limits be taught in a freshman (or high school) calculus class? In particular, what role should two well-known, yet generally disliked, Greek letters play in the beginning weeks/months of the course?

It is surely the case that if one is to do things properly (i.e. rigorously) one must deal with arguments. However, it is also surely the case that if one is to do things properly, one must prove, for example, the extreme value theorem. I think this is a viable way to teach calculus, though I think it unlikely something like this would actually be taught outside of a place like Harvard (I surely don’t think something like this is likely to happen any time soon at the University of Illinois).

The question that remains is what should be done in calculus courses that do not attempt to have a completely rigorous treatment of the subject. I think that what currently is done is very poor. As I have looked at books and tried to see what is taught, it seems that the usual treatment goes something like this:

First: Give an intuitive notion of what a limit is (this might be precisely stated, but it is rarely, if ever, precisely referred to). Work with this a bit to evaluate or “conjecture the value” of some limits.

Second: Formally state the definition of limit. Work with the definition to prove trivial things like .

Third: Work with nontrivial limits, essentially disregarding the definition of limit for the intuitive notion worked with initially.

It seems that by treating limits as such, we do a great disservice to students and give them, in fact, the very opposite impression of mathematics than what we should give them. What they see is that mathematics (notation and arguments) turn something intuitively obvious (e.g. ) into something very difficult (perhaps totally incomprehensible). In fact what mathematics does (or should do) is take very difficult problems and make them comprehensible (mathematical language allows us to precisely talk about something that is very difficult and mathematical arguments allow us to come to correct conclusions about those difficult things). Instead of viewing mathematical notation and arguments as valuable tools, students may very well get the impression that they are something that instead muddy the waters.

So what do we do? It seems to me there are three natural options. The first option has already been mentioned: treat things rigorously. Not only do you state the definition of limits, but you use the definition of limit in explicit nontrivial ways (of course if calculus is taught like this, students should also be required to rigorously do mathematics on their homework, exams, etc.) The option on the other end of the spectrum is to never mention the definition of limit, to work with it as a purely intuitive notion, and to not worry about establishing a solid logical foundation on which calculus may securely rest.

The middle option (which to me seems very plausible at a school like the University of Illinois) does just the opposite of what seems currently prevalent. Whereas now the difficulty of abstraction is fully present and the other difficulties are not present at all, this option would allow any natural difficulties and only deal with a portion of the abstract difficulty. What I propose is this: for any limit that shows up in the course (say ) you should expect students to be able to answer a question like: Find positive real number so that: If then . Of course the analogous thing should be required for limits as goes to infinity.

You may say that it only a small step from this to the full definition, but I think otherwise. I think we should not underestimate the difficulty of successive quantifiers for beginning calculus students. Also, it seems clear that if a student is unable to do a problem like this, being able carefully prove that is rather meaningless.

It seems to me that this would be a good step in the right direction for students. We are clearly moving in the direction of rigor, yet we retain some degree of tangibility. Also, the difficulties that the students will face are meaningful difficulties (they will deal with more than abstractions of trivialities).