When should we get into limits in introductory calculus courses?

Question

All of the calculus textbooks I've used (teaching at community colleges) start with the first chapter covering limits. (Perhaps after a review chapter.) I think this order is wrong.

Historically, Newton and Leibniz thought in terms of fluxions or infinitesimals. It took mathematicians 150 years to develop the logical (epsilon-delta) machinery of limits that puts calculus on a sound logical foundation. (And it turns out that this is not the only way to do so. Non-standard analysis uses infinitesimals in a logically rigorous way.)

The big ideas of calculus are derivatives for rate of change, and integration for areas and volumes. I think the course makes more sense for students if we start with "slopes of curvy lines" and velocity. So I'm bringing in outside material to help them explore the basic concept of the derivative from many perspectives. (Our textbook only has two sections that do this. Most only have one. I spend three weeks on it. Much of my material comes from Boelkins' Active Calculus.)

I give the limit definition of the derivative, but I say that for now we'll think of the limit as meaning that we get infinitely close. After we are done with simple derivatives, I go back to chapter one's limit exercises before doing trig derivatives, because we hit some unusual limits there (e.g. sinx/x).

I'd like to know why textbooks cover limits first. Is it just because the mathematicians writing them are stuck at a higher level, and want to first establish the validity of what will be done later? I don't think that's good pedagogy. I think calculus is beautiful and powerful, and that I can convey this better by waiting to introduce limits after students see why they need them.

Why cover limits first?

Answer 1

I'd like to know why textbooks cover limits first.

I don't think there's any big mystery as to why commercial textbooks tend to be similar. It's a market mechanism known as the network effect, the same mechanism that makes Microsoft Windows so popular. Once people start to see something as a standard, anything different becomes non-viable in the marketplace. Stephen J. Gould wrote a nice essay about this in the field of biology; he uses the example of how biology texts propagate the meme of Lamarckian and Darwinian explanations of the giraffe's neck.

To see that there is no objective reason for doing limits first, we just have to look at some older textbooks. Hutton 1807 uses fluxions. Davies 1836 uses limits, but also connects with infinitesimals. Granville 1904 uses limits. Thompson 1910 uses infinitesimals. Different authors did different things, which is how it should be.

There are also some modern texts that don't start with limits and then move on to derivatives: Marsden 1981, Keisler 1976. At the risk of seeming self-promoting, here's a link to a book that I'm a coauthor of.

There may have been a time ca. 1850-1965 when limits were believed to be the only possible rigorous foundation for calculus. That era ended when NSA essentially vindicated Leibnizian infinitesimals.[Blaszczyk 2012]

Other than the network effect, a second and more cynical explanation would be that a course in calculus is used today as a filter in order to get rid of some of the students seeking a valuable credential. This is the phenomenon of credential creep. For example, I teach quite a few students who want to be physical therapists, and many DPT programs require them to take a year of calculus. Calculus is fundamentally an easy subject, but it can be made much harder by emphasizing epsilontics and by requiring these folks to learn a bag of tricks for integration. I'm not a cynic at heart, but I do have a very hard time coming up with any other explanation for why we require future physical therapists to learn how to do integrals using trig substitutions.

Blaszczyk, Katz, and Sherry, "Ten Misconceptions from the History of Analysis and Their Debunking," 2012, http://arxiv.org/abs/1202.4153

Charles Davies, Elements of the Differential and Integral Calculus (1836), https://archive.org/details/elementsdiffere03davigoog

Granville, Elements of the Differential and Integral Calculus (1904), https://archive.org/details/elementsdiffere01smitgoog

Charles Hutton, A Course of Mathematics: In Two Volumes. For the Use of Academies as Well as Private Tuition (1807), https://archive.org/details/acoursemathemat02huttgoog

Keisler, Elementary Calculus: An Infinitesimal Approach, 1976, http://www.math.wisc.edu/~keisler/calc.html

Marsden, Calculus Unlimited, 1981, http://www.cds.caltech.edu/~marsden/books/Calculus_Unlimited.html

Thompson, Calculus Made Easy, 1910, http://www.gutenberg.org/ebooks/33283

Answer 2

It has become almost a dogma that the math curriculum should teach technical prerequisites to what will be covered later. The consequence is that zillions of high-school students learn algorithms for doing partial-fractions decompositions and will never take later courses in which that is used. In effect the broad public learns that mathematics consists of learning to apply meaningless algorithms.

High-school teachers don't know how to alter the curriculum to something that makes more sense, and math professors don't want to do things for students who are so weak that they think mathematics consists of learning to apply meaningless algorithms, which is exactly the lesson that the high-school curriculum designed by math professors taught them.

The students know (and, as this applies to most students, they are right) that if they just learn the meaningless algorithms they will get good grades, but if they heed any hints that there something else to math besides that, that won't help their grades and takes time away from things that would. The professors fail to realize that what they perceive as stupidity among the students is in fact deliberate strategic stupidity; it is in fact a good strategy for what the students want to do. So both the students and the professors are deliberately stupid.

That is why following conventional curricula merely because they are conventional is contemptible.

Stewart's textbook covers

$\qquad$(1) $\displaystyle\lim_x \frac{f(x)}{g(x)}$ where $f(x),g(x)\text{ (both)} \to0$ and

$\qquad$(2) $\displaystyle \lim_x \frac{f(x)}{g(x)}$ where $g(x)\to\ell\ne0$ and

$\qquad$(3) $\displaystyle\lim_x \frac{f(x)}{g(x)}$ where $f(x),g(x) \text{ (both)}\to\infty$ and

$\qquad$(4) $\displaystyle\lim_x (f(x) - g(x))$ where $f(x),g(x) \text{ (both)} \to\infty$ and

$\qquad$(5) one sided limits of artificial piecewise defined functions and

$\qquad$(6) infinite limits and

$\qquad$(7) various ways in which limits fail to exist

and a lot of other possibilities. Guess what students learn from this? They do not learn that case (1) above plays a central role in differential calculus. They write on the final exam that in a particular problem the limit does not exist because the numerator and denominator both approach $0$, and the fact that the whole of differential calculus could not exist if that were true does not occur to them; after all point (1) above; they do not learn that limits justify formulas about derivatives, since as far as they know, math is dogmatic rather than having justifications. The practice of teaching technical prerequisites first is hardly, if at all, distinguishable from teaching that mathematics is dogmatic. It is wrong.

Probably at least $99.99\%$ of those who would tell you to cover limits first have no reason for saying that except that they've never given the question an instant's thought after observing that the definition of "derivative" in its most usually seen form relies on a concept of limits.

Answer 3

My currently preferred approach is to start the course with an introductory lecture explaining the difference between average velocity over a time interval (something we can always in principle measure using a stopwatch) and instantaneous velocity at a given instant of time (which we do not have a means to measure). This motivates the question of what exactly instantaneous velocity is, using the intuition that most of us have that the notion makes sense. (Of course, that in and of itself is a philosophical issue and one that I want to avoid in the course for the obvious reason of limited time!)

Then I explain that one can approximate instantaneous velocity by measuring the average velocity over smaller and smaller intervals of time centered around the given instant of time. (In fact, I never actually use a centered time interval, but rather use the formulation with the instant of time as one endpoint of the time interval, so that the formulas I write down look like what the definition of derivative will be when we define it officially later on.) Then I say that the instantaneous velocity is the number that we would get in the limit where the time interval became closer and closer to $0$. And I emphasize that, of course, we can never actually make a measurement with a time interval of duration $0$ (both physically because we can't start and stop our stopwatch at the exact same instant of time and mathematically because using the same starting and ending time in the formula for average velocity yields the undefined expression "$0/0$".)

This serves to motivate the introduction of limits, and in fact, in the courses I teach, typically this is more or less the definition of a limit. I use the heuristic and imprecise version of $\lim\limits_{x \rightarrow a} f(x) = L$ means that as $x$ gets closer and closer to $a$ without actually reaching $a$, the values $f(x)$ get closer and closer to $L$.

For me, this seems to strike the right balance of motivating limits and discussing how to compute them in the types of cases that will come up throughout the semester, while still being able to quickly get to derivatives so that most of the semester is spent on the two fundamental notions of derivative and integral and, ultimately, the link between the two provided by the Fundamental Theorem of Calculus.

I strongly prefer to follow pretty closely to the textbook's development in a course such as introductory calculus, where I feel that students are often not sufficiently mature to handle the discrepancy between the textbook presentation and the presentation given in lecture. And they are not equipped to handle the inevitable situation where the textbook has referred to concept X in Section A which appears before Section B and I want to cover the ideas in Section B before covering concept X, but too much of the presentation and choice of problems refer to or are based on understanding concept X.

As for why I prefer to discuss limits before derivatives and integrals (neglecting the obvious cultural bias that that is the way I learned the subject), since I cannot speak for textbook publishers, to me there are two big pictures to take away from calculus. One is the study of change (derivatives) and accumulation (integrals) and the fact that the two problems are intimately related (Fund. Thm. of Calc.). The other is that derivatives and integrals, key ideas for understanding our dynamic natural world, are best understood computationally as the limiting process of simpler calculations that only require basic arithmetic (difference quotients for derivatives, Riemann sums for integrals). It should be noted that the subjective value judgment indicated by the use of the word "best" is not universally agreed upon.

In my opinion, these two big pictures should both be presented and stressed to the students. I find it easiest, logically and pedagogically, to present derivatives and then later integrals to model independent processes and stress the point of view that these are both limiting processes. As such, I need to have discussed what a limit is. After defining integrals as limits of Riemann sums, I point out that the fact that integration, like differentiation, is a limiting process is one of two reasons why the two topics are both covered in a common course. And then I foreshadow what is to come by alluding to the fact that there is a deeper and more important link between these two concepts, which I explain a few lectures later when we are ready to discuss the Fundamental Theorem of Calculus.

Answer 4

Regarding your parenthetical comment "And it turns out that limits are not the only way to do so. Non-standard analysis uses infinitesimals in a logically rigorous way" I would like to comment that the opposition limits vs infinitesimals implied here is not entirely accurate. Limits are present in both approaches; the true opposition is between epsilon-delta methods in the context of an Archimedean continuum, and infinitesimal methods in the context of a Bernoullian (i.e., infinitesimal-enriched, as practiced by Bernoulli) continuum.

In Robinson's framework, the limit is defined in terms of the standard part function. Thus, $\lim_{x\to 0} f(x)$ is the standard part of $f(\epsilon)$ when $\epsilon$ is a nonzero infinitesimal.

Answer 5

I think this book hasn't been mentioned yet:

John C. Sparks: Calculus without Limit - Almost

[Disclaimer: I have it on my bookshelf but have to admit that I haven't read it yet, so I can't really comment on whether it's good or not.]

Answer 6

You ask why to cover limits prior the derivatives when it would be easier to cover derivatives first and limits later; to show why are limits good to know.

Why shall we cover derivatives prior without [anything that uses derivatives]?

I think it is easier to define derivatives when the students are "familiar" with limits. Same for integrals with the knowledge of derivatives. What really matters is HOW does the teacher present the part that is taught. One shall show the students an idea where the course is about to go and why.

I accidentaly passed all the exams in a highschool and thought math is one of the most boring and useless subjects. Man, how wrong I was.

At the university I was lucky to have the teacher I had. He followed the syllabus in terms:

1. Setting up the "vocabulary" (lemma, proof, set,...) and "tools" (proof,...)
2. Operations on sets and basics of boolean logic.
3. Sequence.
4. Limit of a infinite sequence. (What happens at the end of the universe?)
5. Series.
6. Sum of infinite series. (What happens when we try to sum infinite ammount of numbers?)
7. Function.
8. Limits of a function. (What happens at the end? And what happens rally closeto some point?)
9. Derivatives.
10. Integrals.

He taught us the way that it was interesting and fun. I cannot forget his "mutated definitions" that were even in tests. He randomly changed quantificators and relation in a definition and ask us to find a sequence/seires/function that fits in. He forced us to think out of the box and use any tool available.

Once you have students' passion, you can either lead them trough blind phase (limits prior derivatives) and they will follow you or directly to the finish (derivatives) and explain all the tools behind later.

Answer 7

I am not advocating any order but one reason to start with limits first is that the concept of continuity is extremely important and useful and limits are the natural way to introduce it.