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In a freshman calculus course for non-majors;

  • Is it to the benefit of the students to include discussion of epsilons and deltas?
  • To what extent, if any, should they be used? For example, just to prove simple limits, or to prove limit laws, or more?

Some authors jettison the rigorous definition to an appendix and never mention it again, even when discussing limit laws. I know that in such a course many teachers never mention the rigorous definition.

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    $\begingroup$ Books in real analysis such as Rudin or Abbott present the definition for the limit of a sequence before presenting the definition for functions. I think this makes things much easier for students. Would it make sense to talk about sequential limits before functional limits in a calculus course? Of course, sequences are usually not discussed until Calculus II, so this may be difficult to implement. $\endgroup$ – Gamma Function Mar 19 '14 at 1:31
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    $\begingroup$ One of the hidden difficulties of $\epsilon - \delta$ is that most algebra preparation is with equalities, and suddenly you have to handle inequalities, with the somewhat unintuitive absolute value's behaviour thrown in. $\endgroup$ – vonbrand Mar 19 '14 at 3:11
  • $\begingroup$ I never encountered the epsilon-delta definition until upper division courses at college. $\endgroup$ – Tim Seguine Mar 19 '14 at 12:31

11 Answers 11

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For some reason there is a wide spread view that the $\epsilon-\delta$ definition of a limit is an obscure thing, relevant only to mathematicians, and that the only reason to care about them is to make limits ``rigorous''. This could not be further from the truth. $\epsilon-\delta$ analysis is all about learning how to control error in the outputs of a function. This is crucial for applications.

To give an $\epsilon-N$ example, say that you are an engineer who needs to write a numerical integrator to compute the number of calories someone is expending when they use an exercise bike. You want to make sure that the result is accurate to 10 calories per hour of use. The question is: how many subdivisions will I need to use? This is incredibly relevant. The $\epsilon-N$ analysis just takes it one step further: If I ask you to be accurate within $\epsilon$ calories, how many rectangles will you need to use?

This applies to the standard $\epsilon-\delta$ style proofs as well. If $S(r)$ gives the strength of a beam as a function of the radius, many peoples lives might depend on knowing how precisely you must machine the radius to land within an acceptable range of strengths for the output.

For many of our students, the functions they meet in their day to day life will not be given by formula, but by numerical data. At this point all the rules of differentiation and integration go out the window, and the most important thing becomes understanding how good an approximation you are getting.

$\epsilon-\delta$ analysis is hard, but it is not only there to satisfy mathematicians with an insane desire for "rigor": it is the very practical and important skill of learning to bound errors.

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    $\begingroup$ +1 Interesting preservative and an excellent answer. $\endgroup$ – Gamma Function Mar 19 '14 at 1:34
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    $\begingroup$ This is something I can tell my students. Do you think it's useful to get them to do ϵ−δ proofs in their first calculus course? $\endgroup$ – Sue VanHattum Mar 19 '14 at 4:22
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    $\begingroup$ The examples are fun but they don't convince me. Your engineer probably won't get a continuous function, but rather discrete data, and surely too low-quality to get calories to within 10 per hour. Your machinist probably is not aiming for an exact strength, but a minimum strength. I gave an example that I think works better in a previous comment: mathoverflow.net/questions/158145/… $\endgroup$ – user173 Mar 20 '14 at 2:32
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    $\begingroup$ @MattF. The beauty of the engineering example is that if your discrete data arranges itself into large, neat, monotone chunks, then any continuous monotone function passing through those data points will have the error bounds spelled out in my comments above. So the engineer has the inverse problem: given the sampling rate, how accurate can they claim their machine is? As for the machinist, you are right that they only care about minimum strength, which suggests that they should place the topology generated by $(a,\infty)$ on the codomain: i.e. this comment only shows the need to $\endgroup$ – Steven Gubkin Mar 20 '14 at 10:40
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    $\begingroup$ tread in deeper waters to make things mathematically rigorous. Of course, the machinist will probably not have to think in terms of topology, and the engineer will probably not have to think in terms of epsilons and $N$'s. Hopefully just having these scenarios in mind can answer the question "Where would something like this come up in real life?" $\endgroup$ – Steven Gubkin Mar 20 '14 at 10:43
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I teach at a community college, and I love teaching calculus. I hope my course is at least as good as university courses. I think it is a big benefit to introduce them to the deep logic of the epsilon-delta definition of limits, but I do not prove anything with them, and don't think that's very valuable for beginners.

Timing
I think any attention to limits at the beginning of the course is not wise. Limits are strange new creatures for the students, and at the beginning of the course they have no idea why this weirdness is necessary. For most students it will be alienating to spend time there before they've learned about derivatives.

So I start with a full unit on understanding derivatives as rate of change, slope, velocity, ... (Anton doesn't have nearly enough of what I want, so I use lots of material from Boelkins' free text for this part.) We use the regular limit notation, but I just say that it means infinitely close. (And I say that that's not really enough. But it's about what Newton did, and mathematicians took 150 years to do better, so it's good enough for a start.)

Then, after the units on rules of derivatives (I call them properties) and applications, once the students see how valuable derivatives are, I introduce limits more formally. I do not ask any test questions regarding limits, but I do introduce the epsilon-delta definition.

Definition
I call it the sentence that took 150 years to write, and I write it almost like a poem, one short phrase on each line:

lim (x->c) f(x) = L
means
for any ε > 0
there is a δ > 0
such that
if 0 < abs(x-c) < δ
then abs(f(x)-L) < ε.

I draw diagrams with colorful bands, much like what you see in any good textbook, and explain with lots of pointing.

Example
Then I give a colorful example to help the students understand the machinery. (I got it from a colleague many years ago. Thanks, Diane.) It uses a cookie crispness index on the y-axis, and the time your cookies are in the oven on the x-axis. Before I get into it, I ask the students what their favorite cookies are and how crisp they like them on a scale from 0 to 10, where 0 is doughy, 3 is still very soft and chewy, 6 is getting more crispy than chewy, and 9 is starting to get burned. Then I use one student's preference for my example.

I draw a line for the function that connects cooking time to crispness, and from that I estimate the time I need to bake. Then I play with them about how perfect the cookie will be (ε). Now I get to draw my bands. If I want the cookie that close to perfect, how close to perfect must my timing be, when I pull that tray out of the oven?

No testing, no proofs, just a good example to chew on. ;^)

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    $\begingroup$ Just introducing the definition, and never doing aything "for real" with it sounds odd... and counterproductive. $\endgroup$ – vonbrand Mar 19 '14 at 1:19
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    $\begingroup$ I don't believe they're ready to do much with it. What would you suggest? $\endgroup$ – Sue VanHattum Mar 19 '14 at 4:20
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    $\begingroup$ With no testing, do they internalize the logic enough for later use? They might alternatively remember the example, but conclude that mathematicians are averse to putting undercooked cookies back in the oven. $\endgroup$ – user173 Mar 20 '14 at 2:01
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Given that the specific (and not at all unique) "rigorization" of calculus took 200 years (from Fermat, Descartes c. 1650, up to 1850), as a reflection of human perception, it seems to me completely unreasonable to insist that novices "appreciate" the rigorization-throes of the mid-to-late 19th century. (Indeed, for all the citing of K. Weierstrass as exemplar of "rigor", his self-citing absolutely does not emphasize foundational things, but, rather, work on several complex variables, abelian functions, and other very-easily-arguably-good, but not "elementary", topics!!!)

As I have suggested at other moment in similar venues, if we were to pretend that students were critical intellectuals (in the best sense), they'd surely not bend to mere authority or commands to comply, but would ask "what advantage?" (which I would universally encourage).

Then it does become quite difficult to create any imperative to do epsilon-delta... because all the live issues, about modeling mechanics, or even the simplest calculus, do not seem to need confirmation!!!

That is, the two centuries of amazing discoveries enabled by the very idea of calculus were the motive force. "Rigorization" was a luxury, and of interest only to specialists... even though, yes, Cantor was led to Set Theory by worrying about convergence of Fourier series. Cool, yes, and very interesting at some point, but not the critical factor in Calc I.

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  • $\begingroup$ Interesting point about the history. Just to make sure that I understand: Are you saying that one shouldn't do the definition or that one doesn't have to? $\endgroup$ – Thomas Mar 19 '14 at 1:00
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    $\begingroup$ @Thomas, I assert that one should not do the "epsilon-delta definition" for quite some time (months? years?) after one shows all the amazing things one can do with calculus, as Euler and the Bernouillis did in very sophisticated ways many decades before "the real numbers were constructed", and so on. $\endgroup$ – paul garrett Mar 19 '14 at 1:10
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    $\begingroup$ Ok, thanks for clarifying. $\endgroup$ – Thomas Mar 19 '14 at 1:10
  • $\begingroup$ Surely you wouldn't argue that elementary education should start out with Roman numerals or even more archaic numeral systems. $\endgroup$ – Gamma Function Mar 19 '14 at 1:49
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    $\begingroup$ one should not do the "epsilon-delta definition" for quite some time (months? years?) after one shows all the amazing things one can do with calculus I see that in your calculus refresher math.umn.edu/~garrett/calculus you put limits quite early, and you then present the product rule without proof. I realize that the book is intended as a refresher; would this also be your preferred approach for an introduction? It seems to me that if you do limits after developing all the techniques of calculus, then you either can't prove the product rule or must prove it using infinitesimals. $\endgroup$ – Ben Crowell Jun 22 '14 at 23:15
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I've taught calculus with and without $\epsilon-\delta$ techniques. In particular:

  1. at a large state school with a nearly endless supply of engineering students. Here math majors were a rare event... I would characterize the main goal of the course just to get the students up to speed computationally. For many students, they only really learn algebra in calculus. I say learn in the sense that they apply algebra to solve problems which cannot be fit into some standard form. In calculus, the algebra is wild, untamed, it goes where it wishes and you better be able to deal with it. This struggle ideally produces students with greater algebraic maturity and, naturally, a working understanding of the conceptual accomplishments of calculus. Proofs were rarely given and even when they were I would often give a "physicsy proof". For example $\sin(h) /h \approx 1$ is obvious if you just look at a tiny triangle incribed in the unit circle. I suppose my perspective when first teaching was in part the natural outgrowth of my origins story; I was raised by physicsists. So, the question for me in teaching was how can I prepare students to face anything which may be thrown their way in junior/senior level courses? My goal was to prepare them for those trials and tribulations.
  2. at a private liberal arts school. Here, proportionally, I have a nontrivial population of math majors. So, I cannot ignore the need for laying a foundation. I still care about the calculations, computations and applications. But, I make room for proofs. In 2011, I made a major revision to how I teach the course. That summer I wrote course notes which make an effort to prove as much as possible for the first course. Partly, I just want to demystify the idea of a proof for them. I want them to see that a proof is not some rigid construct as we sometimes encounter in highschool geometry. I also want them to see that we can give a nearly complete account for why things are true. Of course, I do not test too heavily on the proof aspect of the course. For example, in treating $\epsilon-\delta$ I only require the linear and quadratic cases for the test. I prove several limit laws from the precise definition. Now, for the majority of students, this is probably annoying. But, honestly, they have the same feeling about limit laws. Every approach has it's advantages and disadvantages and I think what is best depends on your particular audience. That said, I found this approach has attracted students from other majors. There exist students who are looking for something deeper. All of this said, I'd trade the proofs in calculus for a required second course of linear algebra in a heart beat.
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My college first-year calculus class was by a professor who did teach the subject because he loved it. After a clear explanation of what limits are about (sorry, it was 40 years back, no details left by now), he explained the definition, and did a few simple ones, like $\lim_{x \to 2} x$ and $\lim_{x \to 2} x^2$. He made quite a show out of his "secret technique to prove limits" (the typical proof where you are given an $\epsilon$, conjure a complicated formula for $\delta$ out of thin air, substitute and get a clean ${} < \epsilon$ at the end). It was simply to work backwards "on scratch paper, but make sure none of the spies from the parallel classes sees this!!" and the "for clean" retrace of the steps. All seasoned with a few extra tricks, and finally discussion of $\lim_{x \to 0} \frac{\sin x}{x}$.

As other answers say, he soon dismissed this as "reserved just for mathematicians, for you engineers we have simpler tools" and introduced (and proved) the classical theorems on limits. He occasionally went back to the definition (sometimes just as a reminder, sometimes to prove a tricky limit where theorems weren't applicable).

One of the critical things was that he carefully made sure each had understood each step before going ahead. I suspect we didn't cover the complete official contents of the course (did't matter, he was dean of the Science Faculty at the time).

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I have taught calculus several times and I have always included the $\epsilon-\delta$ definition of limit. And I haven't really ever had any problems.

What I have done is to first introduce limits of functions be using graphs. I explain what it intuitively means that the limit of a function exists. I make very clear that the initial presentation of the concept is just to get an intuition.

After that we do a lot of examples using this method. We talk about limits that are pretty obvious and some where rewriting the function makes the existence of the limit clear.

After this, I then state the precise definition. I should the several examples of how to prove that the limit of a function is a given number. I make sure to keep the format of the proofs the same. So I always start by writing "proof" and the "Let $\epsilon > 0 $ be given" and so on.

After showing several examples of even quadratics, I go back and end with some linear functions. For the exams I just require them to know how to prove the limits of linear functions.

And all this in general works out pretty well. I will admit that the first example or two are very confusing for the students, but after realizing that we keep doing the same thing over and over, they come to appreciate the rigor.

Now you could(should?) include examples where you have a specific value of $\epsilon$ given. The students then have to find a/the corresponding value of $\delta$ that will work. In my experience I have done this right after the definition or after you have done all the examples. I am probably leaning towards doing the specific value examples right after the definition as a help to understanding what we are doing.

Anyway, so from my experience it really isn't too hard to teach the $\epsilon-\delta$ definition. And the nice thing is that you can prove some of the rules that we use for limits, like $\lim_{x \to a} c = c$. You might even require your students to memorize some of these simpler proofs.

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I'll probably be downvoted or flamed for this, but after teaching calculus for 20 years both with and without $\epsilon-\delta$ definitions of limits, I've decided that there is a time and place for such treatment of limits, but it is not in an introductory course for non-majors. The payoff is simply not worth the investment at this stage.

What I've experienced is:

  • Students at this level are still developing their mathematical maturity and are not (generally speaking) at a level where $\epsilon-\delta$ stuff truly makes sense. Hence the majority of students simply parrot the instructor without true understanding.
  • Students lock in single-mindedly on these problems as if the whole of their understanding of calculus depended on it, which at this level it does not, and then miss out on understanding concepts that are within their grasp -- like the simple notion that a derivative indicates a rate of change.
  • The overall instructional goals for an introductory course in calculus for nonmajors do not typically encompass understanding of $\epsilon-\delta$ proofs. Put simply, this is not what students in the course are supposed to be getting out of the course, and our client disciplines are telling us this as well. Check out the MAA CRAFTY report for what our client disciplines are asking for and you will not see "a deeper understanding of mathematical proof" among any of the things that are listed.

You can make a calculus course without $\epsilon-\delta$ quite rigorous and demanding, and make it so students come away with a strong understanding of both concepts and computation that they can then take back to their home disciplines and use effectively. This is the main goal of such a course. Adding in $\epsilon-\delta$ in this situation becomes more like one of those memory-hogging apps that runs in the background on your phone or computer -- it takes more away from the course than it adds. Instead, save that stuff for the second time through calculus (= advanced calculus/real analysis).

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Yes, teach the entire proof. Here's why ...

Limits are the underpinning of the entire field of the Calculus of a single variable. They bind the inverse operations of integration and differentiation together through the limit of the difference quotient, and the limit of the Riemann sum, together with the Mean Value theorem for Integrals. Without limits, the Fundamental Theorem[s] cannot be proven. The concept of a limit is used for nearly every convergence and divergence test, L'Hopitals Rule, definitions of asymptotes, the proof that continuous compounding equates to the law of exponential change, required for a rigorous proof of the chain rule, required for proof of all the derivative formulas, required for proof of all of the integration techniques, etc., etc. It literally binds the entire field together.

As an AP Calculus AB and BC teacher in high school, I initially treated the instruction of limits much like it was treated when I was student, namely, rushed over, given only a cursory look, and slandered as 'kid stuff' or 'simplistic,' concepts that were only necessary for the 'big stuff' up ahead. Not surprisingly every time I found myself teaching the later material, I began to realize, as an educator, that every contention I attempted to make about the subtlety of the calculus relied on limits as the underpinning of those arguments. Don't get me wrong ... I had always proved nearly every algorithm in the course, and stressed the importance of epsilon delta proofs, etc., but had skipped over that of limits. I am now firmly of the opinion that we do the students a huge disservice when we do this ...

Consider the beginners understanding of all the concepts in the list I provided above ... and then consider telling the beginner the truth. That you refuse to prove that any of it is true ... Well, that's essentially what you are doing as an educator when you don't demonstrate the epsilon delta proof of limits. Why? Because that proof underpins the entire canon of the calculus. As a student, it left a bad taste in my mouth that I never fully appreciated until I started teaching the course ... To be sure, there are some things that are too difficult to teach first year calculus students, e.g., a rigorous understanding of the ramifications of the implicit function theorem and how that can be extended to the concept and basis of implicit differentiation. That's certainly a topic I forego ... BUT ... limits are the bellybutton of calculus ... teach and demonstrate the full proof, master it to teach it swiftly and with grace, and hopefully students will understand the different phenotypes of limits and have a more robust appreciation of the calculus ... That's what I do now with my BC students and I am starting it with the AB students next year, for what it's worth ...

Regards, Jonathan Haack

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    $\begingroup$ This seems like a weak argument to me, since calculus came about ca. 1690, while limits date to ca. 1820. Something must have been binding the whole field together for over a century, and it wasn't limits. $\endgroup$ – Ben Crowell Jun 22 '14 at 23:38
  • $\begingroup$ Well, I recently became more amenable to your idea, namely, that patterns were found first and then others came and tied everything together with proofs later. It not only makes sense, it is historically accurate. Nevertheless, there is a beauty to proofs in and of themselves, so I guess although I have revised my opinion of the development of mathematics to be in line with what you propose, I suppose I still believe it should be taught in order to see how it was bound together at a later date, as it were. Thanks for your comment. $\endgroup$ – oemb1905 Sep 28 '14 at 8:16
  • $\begingroup$ At the same time, if it was not limits binding calculus together - as you say - then what was it? I would propose, despite the other discussion about proofs versus patterns, that the limits were there all along, but just not discovered. $\endgroup$ – oemb1905 Jan 8 '15 at 2:55
  • $\begingroup$ Additionally, although the language was different ... limits were abound in the Principia. $\endgroup$ – oemb1905 Jan 8 '15 at 2:57
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Based on my own experience and the students I tutored in calculus, it took me several tries to "get" the epsilon-delta definition of a limit, including writing out the definition in formal logical symbolism and figuring out what it meant if the limit did not equal a particular number. The epsilon-delta technique is an inordinately cumbersome method for finding simple limits for simple functions, and the various limit theorems that can be proved with it are simpler and more useful. I was baffled at finding an appropriate delta for non-linear functions, and found no further use for the technique until I took a course in analysis.

For a beginner in a non-rigorous course, I would find it appropriate to introduce limit theorems without formal proof at first, and save the formal definition as an optional advanced topic for those who are doing well with the basics.

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  • $\begingroup$ But why should they believe the limit laws? $\endgroup$ – JP McCarthy Apr 23 '14 at 8:29
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With so many applications to teach, so many numbers to calculate, so many functions to graph, so many applications.... No, I wouldn't find it beneficial to discuss rigorous limits very much; they would be low-priority for me.

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  • $\begingroup$ I believe that this should be a comment. $\endgroup$ – Brian Rushton Mar 20 '14 at 12:22
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    $\begingroup$ @Brian Rushton: Fair point, I added the word "no" to clarify the answer. $\endgroup$ – user173 Mar 20 '14 at 12:26
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To be completely honest, sometimes I wonder why we include limits in the curriculum at all, at least at the place we do.

The traditional place they are found is just before derivatives, which makes mathematical sense, since the definition of derivative is a limit. However, almost all of the students I talk to individually (several hundred a year), keep the ideas of limits and derivatives completely separate in their minds, at best treating limits as an annoying thing they did before they got to the real business of differentiating.

I would much prefer that differentiation was taught as a process for finding a formula for a rate of change of a function at particular points -- a new function where when you sub in the point, it tells you the rate of change. Through much observation our forefathers realised there were all these neat relationships for various functions and combinations thereof, and also that it could be used to solve all sorts of useful problems. In essence you teach differentiation straight up as something that you do to functions. I have no evidence to suggest this is the best approach, but I have much empirical evidence that this is how students already think about it.

I'm not necessarily suggesting to completely remove limits and the definition of derivative from the curriculum (though it probably wouldn't hurt your non-maths majors too much to do so). What I am suggesting is to get them familiar with the mechanics of how they behave and the usefulness of them before getting into the foundations of mathematical rigour it's built on.

I can imagine, after you've gotten them smoothly doing derivatives, that you could have a section of the course where the explicit aim is to lay the mathematical foundation for derivatives and other concepts of function. This is where limits belong -- they belong to the general discussion of continuity and differentiability. From the raw meaning of function, what does it mean to say a function is continuous? What does it mean to say a function has a derivative? What sorts of things can you generally say about functions and their derivatives and when and where they exist? (That is, theorems like the Mean Value Theorem.) These concepts only really make sense to ask about when you have already got a good understanding of the behaviour of functions, and indeed they will probably make more sense because you can use their already-existing intuition of what ought to happen to support their learning of what does happen.

If the purpose was to actually discuss the mathematical foundations -- after having built something that is worthy of having a foundation -- then I don't see a problem using epsilons and deltas. It would seem fitting if we were going to be rigorous to really do it properly. You could choose how quickly you switched from that to the limit laws based on the level of your students, of course.

It would of course be important that any time you did ask them to prove anything, that you were clear on what they were and were not allowed to use -- for example, the epsilon-delta definition or the limit laws; differentiating using the rules or with limits?

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