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 . 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:
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!
 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.