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In a freshman calculus course for non-majors;
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.
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.
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.
I've taught calculus with and without $\varepsilon$-$\delta$ techniques. In particular:
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. ;^)
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).
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.
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:
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).
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
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.
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.
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?
