Almost all calculus classes start with limits. The consensus seems to be that it is impossible to discuss derivatives or integrals without first discussing limits. But of course, calculus without limits is possible: it is just numerical calculus.
Instead of asking “how can you talk about calculus without limits,” this post will argue that we should instead ask “how can we expect students to engage thoughtfully with limits unless they first see a problem where they are needed?”
Starting with numerical calculus lets you begin by hooking students with a lot of really interesting applications, since you can do numerical calculus with any dataset you find interesting. Examples I use (with links to my activities) include COVID, water in a (real) reservoir and the growth rate activity described in more detail in this post. Starting with numerical calculus also lets you give students good intuition for many of the big ideas of calculus, for example the first derivative test and signed area (well illustrated by the reservoir) and concavity (well illustrated by the growth rate example).
Limits are incredibly useful, but they also add a layer of complexity to some intuitively obvious things. Students have absolutely no problem grasping that someone can continue to grow even if their growth rate is slowing: a decreasing growth rate is manifestly different than a shrinking person. But they often have to think hard about how something can be increasing but concave down. Students benefit greatly when they have some common-sense examples in mind before they start thinking more abstractly: increasing but concave down is what happens when someone grows at a at a decreasing rate.
When it comes time to introduce limits, numerical activities can actually give a nice entry point. If you do an activity like this one, in which students do something first with large intervals and then with smaller ones, they will quickly see that smaller intervals lead to a more accurate answer. So, to make calculus as accurate as possible, we want to use the “smallest possible intervals.” for both derivatives and integrals. In a numerical context, there is no question how small the “smallest possible” interval is, but as we transition to continuous functions this language provides a context in which to start discussing limits.
Having done various numerical examples, I introduce calculus with limits by saying to my students:
“Whenever we use a numerical derivative to approximate slope at a point, we want to find the slope between the given point and the nearest other point we have available. I want you to find the slope of \(\scriptsize x^2\) at \(\scriptsize x=3\), using (3,9) and the nearest other point you can.”
First someone might try using x=3.1, but I ask them if there is a closer value to 3. With a bit of nudging, they quickly convince themselves that the slope is 6 at x=3, and that getting closer to 3 makes little difference past a certain point. Without me saying a word to define a limit, they are already using the idea on their own. All of this gives an excellent way of talking about what a limit is: we say that 6 is the slope at x=3 because we may make the slope between two points arbitrarily close to 6, simply by brining the second point close enough to 3. It’s not just that the slope gets close to 6, its that we can make it as close as we could possibly want. Even without an algebraic proof, students are pretty convinced of this.
If you want to scaffold this more, you can use something like the activity outlined in this document, or you can use the idea of local linearity as suggested in this post by the college board.
After having students explore the derivative of \(\scriptsize x^2\) at a specific point, I ask them to try several other points: x=1, x=2, etc. They quickly see that the slope is always double the x-value. Which is a remarkable thing: it brings order to what initially seems to be a bit of a daunting problem. My students are delighted to be able to find the slope, at any point, without having to do another numerical calculation.
In transitioning to a more algebraic notion of a limit (away from a numerical one), we can write an expression for the slope between two points on the graph of \(\scriptsize x^2\) . As long as we use a nonzero \(\scriptsize \Delta x\), the slope between \(\scriptsize x^2\) and \(\scriptsize x+\Delta x\) on the graph of \(\scriptsize x^2\) is $$\frac{(x+\Delta x)^2-x^2}{\Delta x}=2x+\Delta x.$$ Critically, note that this equality does not require a limit. The slope between two distinct points on the graph of \(\scriptsize x^2\) is always \(\scriptsize 2x+\Delta x,\) no matter how big or small the nonzero value of \(\scriptsize\Delta x\) might be.
If students have already done a numerical activity to find the derivative of \(\scriptsize x^2\) , then it makes a lot of sense to do this algebra (without a limit) and then ask “the slope between two points is always \(\scriptsize 2x+\Delta x\). What happens to this expression when \(\scriptsize \Delta x\) gets close to zero?” Answer: the slope gets close to \(\scriptsize 2x\), just like it did in their numerical examples. And, the slope can be made arbitrarily close to \(\scriptsize 2x\) by picking a sufficiently small \(\ \scriptsize \Delta x.\)
Having done all this, students are truly ready to see the definition of a limit: they have everything they need to understand why one would want to take a limit, what it means for an expression to get close to one value as a variable gets close to another value, and why it is important the the expression gets not just close, but arbitrarily close to the stated limit.
If you like this and want a copy of my whole book, please click here