# When should we get into limits in introductory calculus courses?

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?

• On a whim, I thought I might google the phrase Calculus without limits. Indeed, a PDF of potential interest shows up: hawkeyecollege.edu/webres/File/employees/faculty-directory/… – Benjamin Dickman Sep 5 '14 at 16:33
• It's brilliant! I've seen the textbook using infinitesimals, but it didn't seem any easier for students than the limit treatment, and it's not the standard they'll see elsewhere. But this is great! Short enough for interested students to read. And a great way to show that you can "get away with" thinking about infinitely small bits. (There's a great proof of (sinx)' = cosx that uses infinitesimals. Takes it from my four-page explanation to about one page: thephysicsvirtuosi.com/posts/trigonometric-derivatives.html) – Sue VanHattum Sep 5 '14 at 17:13
• It's another case of the "question answered before it is asked" approach to education that plagues our system. For those who think learning is just remembering everything you're told, this makes some sense. For the rest of us who think the students actually have to be engaged in meeting and addressing problems, it does not. – rschwieb Sep 6 '14 at 13:05
• @rschwieb, I'm not sure what 'it' refers to here. My approach, or the text's? – Sue VanHattum Sep 6 '14 at 22:04
• @SueVanHattum I was referring to most run-of-the-mill texts for students at that level and lower. – rschwieb Sep 6 '14 at 23:29

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

• I'll look at your text. I love Boelkins for my first unit. And I mostly use our official text (Anton) for the rest, though I put things in a different order. I'd love to put together my own textbook, though I don't know how many others would use it. – Sue VanHattum Sep 6 '14 at 5:59
• I think your approach is very... american. I was always told that we study for ourselves and not for our job: it is clearly a cultural difference between US and old Europe :-) – Siminore Sep 6 '14 at 10:07
• @Siminore: If a student takes a year of calculus simply because he likes math, that's great, but requiring it is a different matter. Similarly, it's wonderful if a future physical therapist wants to learn Homeric Greek -- but I don't think it should be required. – Ben Crowell Sep 6 '14 at 23:13
• @BenCrowell Actually the university system in my country used to be less focused on future job perspectives than it is in your country. – Siminore Sep 7 '14 at 8:23
• How is assuming calculus is used for weeding out students cynical? That seems like a perfectly sensible use for it. Even with the (extremely miniscule amounts) of epsilon-delta definitions of limits it is, as taught in the US, a fairly easy subject for anyone willing to think a little and do some work. If you do more work you don't really even have to think. Having barrier subjects is quite usual I believe to make sure you don't spend time and money on students who aren't willing to do the work. – DRF May 4 '17 at 7:43

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.

• I tend to agree with your post here. I think the case can be made for the necessity of limits by discussing the speedometer on a car. By discussing how we want to measure rates of change over smaller and smaller durations of time it naturally leads to concept of a limit. Moreover, the fact that it will have the form $0/0$ is manifest. I then tell them that is why we are going to study limits with a focus on tricky ones which have this indeterminant form. I also try to connect it to geometry as many have physics-phobia... – James S. Cook Sep 7 '14 at 2:35
• I agree very much that the derivative should motivate the definition of the limit, and not vice versa, but I would go farther: in an introductory calculus course, why introduce the limit as a separate concept at all? (I don't think 'continuity' is a satisfactory answer; no book I've seen gives any interesting discussion of continuity beyond what can be 'seen' from the intuitive description. Better to differentiate x^2sin(1/x) than to test x sin(1/x) for continuity!) – LSpice Sep 7 '14 at 6:33
• @L Spice I think one good reason to do limits first and discuss continuity is to give the first example of linearity, products, quotients, composites. The inheritance for new functions from old. These are interesting for continuous functions. Also, just to have a language to ubiquitously say when the function is continuous you can just plug in the limit point. It's a baby example of structure and definitions. Is it entertaining? Probably not. – James S. Cook Sep 7 '14 at 17:59
• @JamesS.Cook, differentiable functions also exhibit interesting behaviour under the operations you mention. Basically, I think that continuous but differentiable functions are not interesting objects to study in their own right in a calculus course. Saying that continuous functions are ones for which you can plug in the limit point is unconvincing for someone (like me) who argues that limits as independent objects also do not belong in an introductory calculus course! – LSpice Sep 7 '14 at 23:50
• @LSpice maybe you should write a question which presents your view of what calculus I should cover, perhaps with a sample plan and pointed details of how definitions are altered as to avoid limits etc. It might be productive to understand our difference of opinion. – James S. Cook Sep 8 '14 at 2:52

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.

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.

• I like your precision of language. If you can propose a minor edit to fix this in my post, I'd be grateful. I do want to set up an opposition between the standard (epsilon-delta) limits material and infinitesimals. Mainly because I love this proof: thephysicsvirtuosi.com/posts/trigonometric-derivatives.html – Sue VanHattum Apr 24 '17 at 19:28
• @SueVanHattum, thanks. I will try to make a precise edit to the question. It will have to be approved because I am still below 1000. – Mikhail Katz Apr 25 '17 at 7:31
• I wanted to make less distinction than you do, so I made a smaller edit. Thanks. – Sue VanHattum Apr 26 '17 at 13:42
• @SueVanHattum, no problem. – Mikhail Katz Apr 26 '17 at 13:46

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.]

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.

• Why is continuity important at the beginning of the course? – Sue VanHattum Sep 6 '14 at 22:02
• After asking SueVanHattum's I think very apposite question, I would ask another: who says that limits are the natural way—or, more to the point, more natural for whom? If for students, then I disagree; but if for the instructor, then who cares? – LSpice Sep 7 '14 at 6:35
• @SueVanHattum : it is important at the beginning if you think the concepts of continuity/discontinuity are more important than rate of change (derivatives) or areas (integrals). To apply mathematics to real life (in physics, biology or economics) we have to make lots of simplifications and use approximations. Continuity is crucial to guarantee the approximations still "work" but not all phenomena are continuous. I would like my students to have a critical eye and not assume all functions are continuous. – Sergio Parreiras Sep 7 '14 at 14:24
• @SergioParreiras: If not using limits how would you introduce/teach continuity? The notion of continuity, in some form, predates the notion of a limit by 2000 years, so certainly there are other ways to do it. The traditional informal description is that a function is continuous if you can draw it without picking up your pen. At a more formal level (which is irrelevant to the vast majority of students), there are also multiple ways of defining continuity. See Keisler, math.wisc.edu/~keisler/calc.html , p. 125, for a freshman-level example. In SIA it's an axiom. – Ben Crowell Sep 7 '14 at 15:56
• You write that limits are the natural way to introduce continuity but what evidence do you have for such a hypothesis? The historical evidence points to the contrary conclusion; namely, Cauchy who invented continuity defined it without reference to limits, by requiring that every infinitesimal change in the $x$-variable should always produce an infinitesimal change in the $y$-variable. – Mikhail Katz Apr 23 '17 at 11:53

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.