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?”
Sam Shah’s blog has many interesting posts, and this post is related to Sam’s post on getting rid* of limits. In brief, I don’t think that we should get rid of limits, but I do think that they are a terrible place to start a calculus class, and I also think that they are much less central to calculus than many people seem to think they are.
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.
After doing some numerical examples, I like to introduce the power rule (before formally discussing limits) using a numerical activity like the one described in this post.
In transitioning to a more algebraic notion of a limit (away from a numerical one), I 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 \(\scriptsize x\) 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. After doing all these activities, limits become a vital and important tool, one which addresses a clear need. And students who care about limits will be far more motivated to learn about them.
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