Making epsilon-delta proofs not just precalculus

In trying to find lecture-length videos of epsilon-delta proofs, I've found that almost all of them just start with the definition and then work through the algebra to get the answer. In effect, it reduces the problems to "do you have the pre-calculus algebra to solve the question?" and all the questions are basically the same template-like format but with different numbers.

I'm not sure if viewers would know why the definition is the way it is or why the proof is structured that way.

When teaching $$\epsilon$$-$$\delta$$ proofs, what are you strategies to make sure students truly understand the concepts (beyond mere calculations)? Of course you can't completely escape computation and manipulation, but I don't want my students to be computers who can only do problems they've seen before; I want them to be free-thinkers who can expand beyond what I've shown them.

To give a concrete example of what I'm talking about, in my own video at 25:25, I made a part that shows some creativity in picking the $$\epsilon$$-$$\delta$$ relationship for any line of $$mx+b$$.

Before that section, I proved that you could have $$\frac{\epsilon}{\left|m\right|}=\delta$$ but then you'd need to do another case for $$m=0$$. But if $$\delta=\frac{\epsilon}{\left|m\right|+1}$$ then you wouldn't have to do 2 cases.

To me, something like that shows a little bit of creative thinking can cut down on computation and show that one recognizes that the $$\delta$$ found in the stratch work is just the maximum value, and that smaller (positive) ones can work.

What are some of the creative ways you make $$\epsilon$$-$$\delta$$ proofs more than just computation?

• Welcome to the site! I'm worried that you've made the barrier to answering your question pretty high. Are you expecting some educators to watch this 40 minute video before answering? If not, you should remove the video link and discussion from the question and put it in a comment (or even in an answer -- it's fine to answer your own questions here -- you could say "i tried to deal with the issue at 25:25 in this video by saying X Y Z"). – Chris Cunningham Feb 18 '20 at 17:57
• @ChrisCunningham Thanks for the feedback for how I'm posing the question. I've reworked the question – Robbie_P Feb 18 '20 at 20:42
• Maybe, in the "algebraically" part of your exposition, you should calculate $f(1\pm\delta)$ instead of $f(1)$, otherwise (at this point) students may think that this value matters (which is not the case). – Pedro Feb 26 '20 at 4:03
• In order to help the understanding, a historical background can be useful (see the paper Who Gave You the Epsilon? by Judith V. Grabiner) – Pedro Feb 26 '20 at 4:13