Mathing…

@Matthew Leingang
You are right.

The comic is about a giving “a practical understanding” of limits. As a high school math teacher, the biggest issue for me is to get students to understand the difference between evaluating a function at c and finding the limit as x approaches c of f(x).

A student first encountering limits will see no difference between limit as x approach c of f(x) and f(c). Very often in textbooks, we start off with examples where there is no difference between the two. Given f(x)=2x+1, find f(3) and find the limit as x->3 of f(x). We also give examples that supposedly clarifies the difference like f(x)=(x^2-4)/(x-2). There is a hole/gap in that function when x=2, but students treat it by mechanically factoring and simply cancelling out the factors of x-2 without regard to what that does to the function. They evaluate using f(x)=x+2, after they cancel out the x-2 terms in the numerator and denominator, for both limit as x->2 of f(x) and for f(2). Students could mechanically get the correct answer and still have no understanding of the difference.

Is this a problem of math teachers not explaining or going over examples over the distinction? I’m not sure, but what I find for myself is that even though I cover the examples, I would say maybe a quarter of the students will really understand the difference if I’m lucky. The rest are still struggling with the notation of limits and trying to adapt new information to something they know from algebra 2. The easiest thing that students resort to is to treat limits as just factor->simplify->evaluate and the distinction is lost.

You are right in that with this model it is not possible to get arbitrarily close to a. In the a future comic I will be using 1000, 2000, 3000 as the position numbers and show if we are unsure about the limit at 5000 we could look at 4000 and 6000. Then we will look at 4900 and 5100, then 4990 and 5010, and finally at 4999 and 5001. It then stops there. However on the real coordinate plane we can go further with examples that show how it’s possible to get arbitrarily closer.

Frank Lee :

okay, it’s a good effort at simplifying limits; it’s a model, and it’s granular; so i recommend that you declare the premise up front and tell the reader what you will set out to demonstrate with the model.
remember:
a useful model simplifies amplifying one dimension of the inscritible,
the model is not overly complex; otherwise, it defeats the purpose (a bellybutton should not be bigger than the stomach – a Korean adage muttered under the breaths of denture toting grandmas),
the model does not have distortions from important underlying fundational elements which would render it unusable,
moreover, does it obfuscate? or ellucidate?

so where does this model stand mr. ho?

Good Point! I’ll update the comic to reflect the goal of the model. Being upfront about the limitations of the model wouldn’t hurt either. Not sure how to go about it, but I’ll give it a shot.

I thought I was elucidating the difference but without explicitly stating its limitations I may mislead students in the long run. I find that transitioning from a discrete model to a continuous model is easier than fixing a misconception about limit as x approach c of f(x) and f(c). Gonna hafta rework the sequence of the next few comics.