I’ve been meaning to write this post for a while. I teach non-AP Calculus. My goal in this course is to get my kids to understand calculus with depth — that means **my primary focus is on conceptual understanding**, where facility with fancy-algebra things is secondary. Now don’t go thinking my kids come out of calculus not knowing how to do *real* calculus. They do. It is just that I pare things down so that they don’t have to find the derivatives of things like . Why? Because even though I could teach them that (and I have in the past), I would rather spend my time doing less work on moving through algebraic hoops, and more work on deep conceptual understanding.

Everything I do in my course aims for this. Sometimes I succeed. Sometimes I fail. But I don’t lose sight of my goal.

Each year, I have parts of the calculus curriculum I rethink, or have insights on. In the past few years, I’ve done a lot of thinking about ** limits **and where they fit in the big picture of things. Each year, they lose more and more value in my mind. I used to spend a quarter of a year on them. In more recent years, I spent maybe a sixth of a year on them.

**And this year, I’ve reduced the time I spend on limits to about 5 minutes.***

*Okay, not really. But kinda. I’ll explain.

First I’ll explain my reasoning behind this decision. Then I’ll explain how I did it.

**Reasoning**** Behind My Decision to Eliminate Limits**

For me, calculus has two major parts: the idea of the derivative, and the idea of the integral.

Limits show up in both [1]. But where do they show up in derivatives?

- when you use the formal definition of the derivative

and… that’s pretty much it. And where do they show up in integrals?

- when you say you are taking the sum of an infinite sum of infinitely thin rectangles

and… that’s pretty much it. I figure if that’s all I need limits for, I can target how I introduce and use limits to really focus on those things. Do I really need them to understand limits at infinity of rational functions? Or limits of piecewise functions? Or limits of things like as ?

Nope. And this way I’m not wasting a whole quarter (or even half a quarter) with such a simple idea. All I really need — at least for derivatives — is how to find the limit as one single variable goes to 0. C’est tout!

**How I did it**

This was our trajectory:

(1) Students talked about average rate of change.

(2) Students talked about the *idea* of instantaneous rate of change. They saw it was problematic, because how can something be changing *at an instant*? If you say you’re travelling “58 mph at 2:03pm,” what exactly does that mean? There is no time interval for this 58mph to pop out of, since we’re talking about an instant, a single moment in time (of 2:03pm). So we problematized the idea of instantaneous rate of change. But we also recognized that we understand that instantaneous rates of change do exist, because we believe our speedometers in our car which say 60mph. So we have something that feels philosophically impossible but in our guts and everyday experience feels right. Good. We have a problem we need to resolve. What might an instantaneous rate of change mean? Is it an oxymoron to have a *rate of change* at a *instant*?

(3) Students came to understand that we could *approximate *the instantaneous rate of change by taking the slope of two points *really really really* close to each other on a function. And the closer that we got, the better our approximation was. (Understanding why we got a better and better approximation was quite hard conceptual work.) Similarly students began to recognize graphically that the slope of two points really close to each other is actually almost the slope of the tangent line to the function.

(4) Now we wanted to know if we could make things exact. We knew we could make things exact if we could bring the two points *infinitely *close to each other. But each time we tried that, we got either got two points pretty close to each other or the two points lay directly on top of each other (and you can’t find the slope between a point and itself). So still we have a problem.

And this is where I introduced the idea of introducing a new variable, and eventually, limits.

We encountered the question: “what is the exact instantaneous rate of change for at ?

We started by picking two points close to each other: and

This was the hardest thing for students to understand. Why would we introduce this extra variable . But we talked about how wasn’t a good second point, and how also wasn’t a good second point. But if they trusted me on using this variable thingie, they will see how our problems would be resolved.

We then found the average rate of change between the two points, recognizing that the second point could be really faraway from the first point if were a large positive or negative number… or close to the first point if were close to 0.

Yes, students had to first understand that could be *any* number. And they had to come to the understanding that represented where the second point was in relation to the first point (more specifically: how far horizontally the second point was from the first point).

And so we found the average rate of change between the two points to be:

We then said: how can we make this exact? How can we bring the two points infinitely close to each other? Ahhh, yes, by letting get infinitely close to 0.

And so I introduce the idea of the limit as such:

If I have ,** it means what blah gets infinitely close to if gets infinitely close to 0 but is not equal to 0. That last part is key.** And honestly, that’s pretty much the entirety of my explanation about limits. So that’s the 5 minutes I spend talking about limits.

So to find the instantaneous rate of change, we simply have:

This is simply the slope between two points which have been brought *infinitely* close together. Yes, that’s what limits do for you.

And then we simplify:

Now because we know that is *close *to 0, but *not equal to 0*, we can say with confidence that . Thus we can say:

And now as goes to 0, we see that gets infinitely close to 6.

Done. (Here’s a do now I did in class.)

We did this again and again to find the instantaneous rate of change of various functions at a points. For examples, functions like:

at

at

at

For these, the algebra got more gross, but the idea and the reasoning was the same in every problem. Notice to do all of these, you don’t need any more knowledge of limits than what I outlined above with that single example. You need to know why you can “remove” the (why it is allowed to be “cancelled” out), and then what happens as goes to 0. **That’s all. **

Yup, again, notice I only needed to rely on this very basic understanding of limits to solve these three problems algebraically: **means what blah gets infinitely close to if gets infinitely close to 0 but is not equal to 0. **

(5) Eventually we generalize to find the instantaneous rate of change *at any point, *using the exact same process and understanding. At this point, the only difference is that the algebra gets slightly more challenging to keep track of. But not really that much more challenging.

(6) Finally, waaaay at the end, I say: “Surprise! The instantaneous rate of change has a fancy calculus word — *derivative.*”

Apologies in advance if any of this was unclear. I feel I didn’t explain thing as well as I could have. I also want to point out that **I understand if you don’t agree with this approach.** We all have different thoughts about what we find important and why. I can (and in fact, in the past, I have) made the case that going into depth into limits is of critical importance. I personally just don’t see things the same way anymore.

Now I should also say that there have been a few downsides to this approach, but on the whole it’s been working well for me so far. I would elaborate on the downsides but right now I’m just too exhausted. Night night!

[1] Okay, I should also note that limits show up in the definition for continuity. But since in my course I don’t really focus on “ugly” functions, I haven’t seen the need to really spend time on the idea of continuity except in the conceptual sense. Yes, I can ask my kids to draw the derivative of and they will be able to. They will see there is a jump at . I don’t need more than that.

In my college classes, I do some work with the formal definition of limit as we’re starting the anti-derivatives unit. They’ve seen (through lots and lots of work like what you describe above) the value of the limit concept, so they’re ready to put some intellectual energy into understanding the formal definition by this time. I think it works well. I want a textbook that does it our way, so they don’t have to jump around so much to find their homework problems.

We should write a book! (Okay, I should be clear I’m not being serious as I barely have time to do my laundry… but we can dream, right?)

I want to write a calculus textbook. But what I have in mind would need a few other names on it, like yours, Bowman’s, and maybe Sean’s. And then the names of some folks who already have creative commons stuff out, that I want to include – but in the order I like! I have so many ideas for it. Can’t start until the first book is out, though…

Check out the Cartoon Guide to Calculus by Larry Gonick. It’s not heavy enough to use a primary text, but makes an excellent supplementary text.

Thanks Nick! I’ll look into getting it. I’m sure I’ve seen it before (I loved another one of his cartoon guides…) but I don’t think I own a copy. Strange!

I found this really interesting. At my college the math dept and the physics dept (I’m in the latter) often argue about what things should get more attention in our respective classes. Limits in Calc 1 come up a lot. One of the most common things that gets brought up is similar to your last point: in physics you don’t tend to run into very ugly functions – most (all?) are smooth, continuous, etc.

I suspect if I were to teach a rigorous calc course, I would probably not get rid of limits as whole heartedly as I did for this conceptual calc course. The reason is precisely what you hit upon? I would make it a point to see where everything we are learning falls apart! And things (like the derivative at cusps) fall apart for not-nice functions!

I didn’t deal much with that sort of thing. It seems clear visually. It would be good to know of an example of a function where not being precise about the limits could mess you up.

samjshah: I think that while limits are of questionable practical value, they represent significant philosophical (and cultural) shift. In my mind, they are what separates us from the classical era. The idea that we can use infinity as a mathematical object is profound (and a bit disturbing). To gloss over it is it abandon philosophy and beauty for practicality.

suevanhattum: I don’t love the idea of relying on visual clarity in calculus. For starters, I (and most students I know) don’t graph every function they’re differentiating. Also, I think handwaving like visual inspection gives students the wrong idea about what math is and may steer them wrong in the future. In particular, in sections about the Mean Value Theorem, students should know that they can’t just apply the theorem willy-nilly; there are times when MVT doesn’t apply.

And, while I’m not sure what might count as function that could “mess you up,” I think f(x) = exp(sqrt(x*x)) might be problematic. And the graph looks harmless enough.

Yes, physicists and engineers mostly only work with analytic functions and don’t see what the fuss is about. Of course, in signal processing (usually taught in electrical engineering) there is the step function and its derivative, Dirac’s delta, which isn’t properly a function at all, though it can be viewed as either the limit of a family of functions (which gets messy) or as a measure, which turns out to justify the way that engineers treat it as if it were a function.

Limits are very important for the classical proofs, not so important for day-to-day use of calculus. There are even theoretical tools now that can make the intuitive notion of infinitesimals (the original way of doing calculus, before limits were introduced) rigorous. The machinery for doing the proofs is a lot more complicated than limits, but the application of infinitesimals easier.

Whether you spend a lot of time on limits depends on what your goal is in teaching calculus—the very useful tool or the example of proof techniques in a fairly simple framework. I can remember my “honors” calculus in college, where we spent a couple of weeks doing all the limit-theorem homework exercises with delta-epsilon proofs. The goal wasn’t to get to get us to apply the limit theorems, but to understand how to construct difficult proofs.

Yes, when I first learned about the hyperreal numbers (even the mere idea of them, and how they actually can be rigorous!!!), I was blown away. I always wondered if introducing the idea of the hyperreals and teaching calculus that way wouldn’t make all the things make more sense? There’s a textbook I remember seeing (maybe I have it?) that does this I think!

I agree re: delta-epsilon proofs. I think they are a good way to teach abstraction, and also the logic behind a proof. Not for my kids, but I remember learning them in high school and thinking they were challenging. And that was a good thing, because math in high school for me wasn’t especially challenging!

Sam, you may be thinking of Jerome Keisler’s textbook. It’s out of print but freely available on his website. http://www.math.wisc.edu/~keisler/calc.html

Yes, I think the Kessler book is exactly what I was thinking of. Thanks!

This is great. Even for a class that rigorously defines limits, going through everything here first as motivation would make a lot of sense.

It seems to me that the “right” answer for any class is to thoughtfully tailor both the focus and level of rigor to the audience and purpose of the class. This is why epsilon-delta definitions are either mentioned only in passing or not at all in a typical AP calculus course, but why they might be among the most important ideas in a college-level honors calc course (or analysis) for the reasons mentioned by Nick and gasstation. On my mind: What role does (non-AP) calculus have at your school? Is it more important to understand calc or to be able to use it? Is it like driving a car vs. understanding how the engine works? Or (as others have suggested), is the beauty and value in the machinery itself? I’d also be interested in what kinds of math, science, and engineering courses your students take after calc.

At the very least, your approach helps to motivate the need for some way to speak about the infinitely large and infinitessimally small in useful, precise terms. Under these circumstances, I can imagine the limit seeming like a useful tool rather than the undermotivated and clunky object it can easily become in calc A/1.

I admit to being a little bit suspicious about the rush to calculus in high school, especially with so many beautiful, challenging, and worthwhile topics to be found “before” calc. But I’ve always been the type more interested in the “why” than the “how can this be used” questions in math. As always, I applaud you for having the courage to tinker (and blog)!

Thanks Sam. I know *why* there is the rush to calculus (well, there are many reasons)… but I don’t think it’s developmentally appropriate for all my kids. Some of them would be better off with a less abstract course.

I assume my kids won’t be taking any more math after this, or if they do take another math, they will be taking calculus in college. At least that’s what I know from kids who come back to visit! Since I know what college calculus is like in my colleges, I hope to give my kids the conceptual foundation to understand all the algebraic crazyness they will be doing if they take it again. My goal is to show them conceptual depth. I wish I could say I also hope to show them beauty. But even though that’s what I think I’m doing with this conceptual depth, I don’t hear many oooohs and aaaaahs. So I’m not doing a great job with that. I think that Mimi Yang [http://untilnextstop.blogspot.com/] has been doing this really really well.

“Conceptual depth.” Love that idea. Too often conceptual is used as a euphemism for superficial.

As I am reading this post I would say that Sam is not abandoning the concept of a limit at all. It sounds as if he is simply saying that all of the algebraic heartbreak that is commonly associated with the study of limits is what can go away (at least in this level of a Calculus class.) Sam, please correct me if I am misreading you.

What I LOVE about your algebra is such a simple, but beautiful step. The line where you have an h/h factored out is fantastic and I am mad at myself for not doing this myself. The whole handwaving of ‘canceling’ the division by zero process has always been a hangup for many of my students. This should go a long way toward eliminating that concern.

I shared this with the Honors Calculus teacher at our school to log away for future planning ideas.

The general idea of a limit is important. Being able to reason out some non-obvious limits (like instantaneous rates of change) is important. Is the notation really that important if you are not going to define a limit (epsilon-delta)?

Sometimes it seems like a lot of “limit” problems are really “limit notation” problems. The value of the limit is either obvious or kind of an afterthought after a bunch of crunching. A “limit” problem would be more about proving the value of a limit or something where it is not so clear how to figure out the value of the limit.

Yup.

I’m a physicist. We most definitely do use delta functions. In fact, we have the audacity to call it the Dirac delta function, after Paul Dirac, the physicist who used the delta function in his development of quantum mechanics. Delta functions can be traced back to Kirchoff, 1882.

More to the point, this post seems to be trying to discover what H. Jerome Keisler, a professor emeritus of mathematics at UW Madison calls “hyperreal numbers.” According to Keisler, this approach of using infinitesimals was invented by Abraham Robinson in 1960. In fact, you can download Keisler’s calculus book, which uses hyperreal numbers, here: http://www.math.wisc.edu/~keisler/calc.html. It develops the calculus without limits.

I couldn’t agree more!! I’ve been teaching calc for about 6 of my 13 teaching years… I think the conceptual is much more important-last year the other calc teacher and I took this very same approach and here is what happened to us -when all was said and done and we tested we had the students graph the average rate of change function over an interval say[x,5]which would have a hole in the graph at the x-value of 5…when we asked the question about the limit as x->5 they would all say that the limit is shown graphically at the discontinuity-it was as if they thought that there is only one possible limit and there would have to be a discontinuity -this was most likely a misconception that I unintentionally created but it was frustrating because we really thought we did a good job and did enough formative assessment to back that up.. So this year we spent time on limits , got bogged down in the algebra and now are struggling to get to integrals and are even more frustrated than before..I hope this was understandable-love your blog and I agree with u !!!

This was understandable! I would suggest that even when you taught limits, the students probably wouldn’t have still gotten the answer to the question you posed… would they? It’s a very tricky concept, that hole. (I have often said to myself how all of differential calculus simply comes from a deep understanding of holes.) But I feel like if we keep on refining how we deal with calculus without all the formalism around limits, we’ll find an even better solution…

There is the first and only rigorous formulation of calculus in human history that is based on analytic geometry:

Part 1: Derivative

Part 2: Integral