Today in one of my two calculus classes today, we got on the topic of .
I think it came up when we were talking about how to approximate the instantaneous rate of change in a problem: we had a function , and we wanted to estimate the instantaneous rate of change when .
So a student said let’s pick another point, such as . And we found an approximate instantaneous rate of change. Of course I asked “how could I get this answer more precise?” and someone said “add more 9s!”
So we realized we could pick a closer point, such as .
Of course, then we had some precocious youngster say: “why not get it super duper exact and plug in ?”
Ah hah. Many of them thought that was SUPER close to, but definitely not equal to, 300.
I went through the whole standard argument, which usually convinces most kids:
Let . Then .
So Which means . Which means .
I thought I had them. One student said I was breaking her worldview.
But then, THEN, they asked me an awesome question.
One said, and the others jeered: “Isn’t kinda breaking the rules of what you’ve been saying. How infinity is a concept? How this decimal goes on forever? And you said we couldn’t mix concepts with numbers. We can’t write or because we’re mixing concepts and numbers. So why can we talk about a number with a decimal that goes on FOREVER? Aren’t we mixing concepts and numbers? Isn’t this thing totally nonsensical?”
Okay, okay, they got me there. And they’re thinking deeply. And they’re getting me to think deeply.
So then I said: “okay, you have a point. So let’s see if we can mathematize this in a way that works with our understanding of things.” So I made a list:
But then I noticed we are still mixing infinity (as a concept) with these numbers because we’re adding an infinite number of them… so then below it I wrote:
Now we’ve written this decimal as what it truly is: a limit as the number of terms gets large without bound. And the limit has a definite value — the sum is getting closer and closer to 1, as close as we want to 1, infinitesimally close to 1, so we can conclude the limit of the sum is 1. So we can conclude that this sum is 300.
And then, sadly, I moved on.
The kids were interested in this conversation, and I think it could get at the heart of what we’re doing with limits (and then relating it to derivatives), and how infinity is a concept (for us) and not a number. But I don’t know what to do from here, where to go from here.
I didn’t convince everyone, and I don’t want to go too far afield with this unless someone out there can suggest a good idea. I mean, this idea of the limit, two things infinitesimally close together, is powerful . So is there a way to extend this discussion meaningfully? Philosophically? Anyone have any good activities out there, any good worksheets out there, any good readings out there, any good videos out there? I’m not even sure what sort of end goal I have. Just something that acknowledges the weirdness of repeating decimals, relating them to limits, and the concept of infinity…
For context for the class, this is the non-AP calculus class, where my kids are at very different levels of understanding.
 From my historical understanding, both Leibniz and Newton (and their followers) were still plagued with the idea that you would be kinda-ish dividing by zero when calculating the derivative, because they didn’t have the concept of limits in their formulation of calculus. This division by zero was unsettling for a number of contemporaries. And it wasn’t until Cauchy came along with his limit concept that he was able to give derivatives a solid philosophical foundation to rest upon.