Delta-Epsilon Magic

In my second year of undergrad I had a formative experience with Delta-Epsilon proofs that stuck with me for a long time. Last week I was able to provide a similar experience for some first year calculus students.

My Experience

At the beginning of class the students had handed in their assignments for the week and the instructor (Ross Stokke) began lecturing; it started with a proof of a limit using the Delta-Epsilon definition. Ross gives some explanation and motivation, and then the TA walks in, ready to collect the assignments for grading.

On the board, Ross begins the proof as always: Let \epsilon > 0 . Of course the next step is to provide a \delta>0 . (Normally you find this by doing a couple minutes of calculation on the back of your page. Students know how to do this, but it takes some time.)

Ross asks the class if they know what \delta should be. We all shrug.

Ross then turns to the distracted TA who is collecting the assignments and asks him what the \delta should be. He looks up at the limit and says “\frac{\epsilon}{3} ” and goes back to collecting assignments. Ross confirmed this was a good choice and continued on with the proof.

We were all flabbergasted. How did he know?

Now I know he isn’t actually a wizard…

I was shocked, but I knew that the TA wasn’t some kind of super-genius. Rather than be dismayed by this impressive feat I was motivated to learn this skill for myself. Suddenly, mastering these types of problems wasn’t something that only super-smart brainiacs could do.

This was an important moment in my undergrad. It jarred me into the mindset that I too could be the master of a skill.

History repeats itself

Flashforward to this school year and one of the calculus instructors (Alfonso) is holding a massive office hour in the Huron lounge for about 25 students (I’m working on a research problem with my colleague Ivan, who is a calculus instructor this year). After about 90 minutes of Alfonso working through problems on the board, the students want to see a “piecewise limit”. So he invents this one:

f(x)=

  • 2x - 3 if 3 \leq x
  • 6-x if x < 3 .

He starts writing the proof (“Let \epsilon>0 “) and I interrupt him.

I ask if any of his students know what the \delta should be. He asks and they shrug their shoulders.

So I yell over to Ivan (who has been helping a student, and hasn’t been listening at all) and ask him what the delta should be. He snaps out of his discussion, looks up, pauses and says “\frac{\epsilon}{2}.”

Alfonso confirms that this will work, and goes on to write the details.

Is this a good teaching strategy?

On the surface this “magic” trick seems like one that should be used judiciously and infrequently. I want math to seem like something we can all do (given enough time, effort and struggle), and this trick seems to go against that principle. This trick could possibly serve to make students feel more unable to do math by pointing out how much better the instructors are at calculus than them.

On the other hand, I like that as a student this trick feels like something you can’t do at the moment but could possibly do in the future. I feel like this is the correct use of this kind of trick; show them something cool, then give them the tools to get to that level.

What do you think?

3 thoughts on “Delta-Epsilon Magic”

  1. Oh, I had a teacher like that in my freshman year. He taught me Calculus II. Me and my friends called it “epsilon magic” or “epsilon juggling”. He would always pick the values to be some strange fractions or other relation functions. And at first it seemed a very peculiar value to choose. But me and my friends (who were pure math undergrads) caught on to this after a couple of times — you just solve the problem with \epsilon , and then you modify the starting value so you’ll have a nice round result at the end.

    I was never a fan of this approach. Not because it obscured something from me, or because it made me feel less capable doing mathematics (calculus had enough effect of that already… and look at me now, a set theorist!). I do not like this approach because it hinders the mind of the student. Especially at first, the focus goes from “how to do this” to “why this value?” which is not a good shift of focus, especially at first.

    Far better, I think, is to engrave into the student’s mind that any function of \epsilon which approaches to 0 would work just fine for these definitions, and after a few times of ending up with a gory result like 2\sqrt{\frac{\epsilon^3}5} , start using this trick. When the students understand why this magical coefficient was chosen, or whatnot. When the student understands the general reason for choosing these weird values, then it’s fine to do it.

    (My apologies in advance if there are typos, or bad MathJax code. It turned out to be quite a long comment.)

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  2. It’s interesting that that was such a powerful experience for you. When I’m teaching delta-epsilonics in my classes I’m never sure whether to make a big deal about getting exactly epsilon at the end (and if I do that, should I choose delta right in the beginning or figure it out at the end), or if I should emphasize that any function of epsilon that goes to 0 is good enough, like Asaf says. I kind of alternate between various ways of doing it because I want students to know that if they had to get epsilon at the end they could, but it’s not nearly as important as whatever they did to bound different parts of the function by something involving epsilon.
    Also, I wonder how many students would be inspired by this like you and how many find it intimidating or off-putting. Thanks for sharing and giving me something to think about.

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    1. My current (untested, possibly awful) hypothesis is that we should abolish the notion that “Delta-Epsilon problems have two distinct stages: Rough Work and Proof” and just stick to the “proof” part.

      I think this is only really an issue when students have a weak grasp of what constitutes a proof. I’ve had many students on tests fill a whole back page with work, then on their test page write a very neat and beautiful solution that starts with “Rough Work: (blah blah blah)” then “Proof: (blah blah blah)”.

      Of course, how do you find the delta? Here we start the proof by establishing a relationship between \vert f(x) - L \vert and \vert x-a \vert . In particular, we always want \displaystyle  \vert f(x) - L \vert \leq \vert x-a \vert \vert \text{SOMETHING} \vert

      Then we should be able to “read off the delta”. This will work for many examples, although becomes a little trickier with some functions like quadratics.

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