The shape in question was , and when I claimed its area was finite, my students didn’t believe me. Actually, my students didn’t even know what I *meant*.

Back when my improv was as good as my planning, I used to love digressing into topics that I found especially interesting, and I indulged myself again this week by spending 10 minutes talking about the idea that a number that never stops getting bigger might have a limit. The idea I focused on this week to exemplify my point was the difference between 1 and 0.99999… “repeating.” I showed, through long subtraction, that there was no difference between the two numbers (the two representations of the same number?). I went to the “ends” of 1.0000… and 0.9999… and did the 1st-grade trick of borrowing from a more significant digit. When my students said, “but you’re still left with a 0.00..01!” I said, “well, we gave up too early!” and added a few digits. It was this last bit that really helped them understand, and I saw that wave of comprehension go across their faces (all of them, I think!).

Since my calculus class is not an AP class, I struggle with the decision to include limits at all. You don’t need them to explain the other concepts of calculus, and since I’m definitely not going to get all the way down to basic limit proofs it makes sense to me to skip them altogether. This breakthrough is a big one, though. If I can create a lesson that leads students to this thought process… I’m in!

If you don’t want to cover it in depth mathematically, maybe do it from a historical perspective.

IIRC, limits were invented to plug a hole in calculus; the difference quotient is always undefined (indeterminate form) at the point of tangency so how do we know the value of its derivative at that point (ie slope of tangent line)?

For most people, canceling out terms so that the difference quotient no longer has a indeterminate form was sufficient. It was good enough that it just worked.

I still think there is value in understanding the concept behind it. It’s how continuity is defined in Calc.

You might try just giving them a practical understanding on how to evaluate limits. This might give you some ideas.

Thanks for the link –

I agree that there is value in understanding limits. What I’m not sure of is that said value is equal to the cost of the class time required to give my students this understanding. For now, I’m content to gloss over as much as they’re willing to let go, and to go deeper only with those students who protest when I say things like “consider an infinitely small interval.” Some limits are easy to accept, like (2x+1)/(2x) as x goes to infinity, and others are harder, like (sin x)/x as x goes to zero. Sometimes explaining an easy one is good enough for even a demanding student, but sometimes they want the whole proof down to deltas and epsilons!