Differing Choices of $\delta$ in a Limit

Question

In conceptually motivating the $\epsilon-\delta$ definition and proof of a limit, I realized a new way of choosing the $\delta$.

For example, consider $\lim_{x\to 4}\sqrt{x}=2$. In the "standard method", you use the back-of-the-envelope calculation that leads to $$\frac{|x-4|}{\sqrt{x}+2} < \frac{|x-4|}{2}$$ and so choose $\delta = 2 \epsilon$. But this leap (replacing $\sqrt{x}+2$ with $2$) seems so unnatural to students who are encountering this for the first time.

Instead, if you draw the conceptual picture of what's going on (which many of us I assume start with), then we get that for $\sqrt{x}$ to be between $2-\epsilon$ and $2+\epsilon$, it must be that $x$ is between $(2-\epsilon)^2 = 4-4\epsilon+\epsilon^2$ and $(2+\epsilon)^2 = 4+4\epsilon+\epsilon^2$. Then, like we do with all the preceding numerical examples, we take the smaller of the two differences of these with $4$ and get $\delta = 4\epsilon - \epsilon^2$.

The resulting proof looks like this:

Let $\epsilon>0$ and set $\delta = 4\epsilon - \epsilon^2$. Then $|x-4| < \delta$ implies $$4-4\epsilon + \epsilon^2= 4-\delta < x < 4+\delta = 4+ 4\epsilon - \epsilon^2 < 4+ 4\epsilon + \epsilon^2.$$ Hence $$(2-\epsilon)^2 < x < (2+\epsilon)^2$$ and thus $2-\epsilon < \sqrt{x} < 2+ \epsilon$ or $|\sqrt{x}-2| < \epsilon.$

This is totally analogous to the argument we expect them to make with numerical examples. So here's my question: Why don't we teach it this way? Why isn't this the way it's done/introduced in textbooks?

Some possibilities:

1. This only works when the function has an obvious inverse. So it wouldn't work for, say, $\lim_{x \to 1} x^5-3x+4 = 2$.

2. Technically if $\epsilon \ge 4$, then $\delta = 4\epsilon - \epsilon^2 \le 0$, and so really we have to break it up into pedantic cases.

So maybe I've already answered my own question, but then the question becomes: Is there any educational benefit to using/showing this type of reasoning as opposed to the usual? Has anyone else done this in class and did it help or hinder your students understanding/proving?

Answer 1

You are answering your own question. But also keep in mind a bigger picture. The purpose of introducing the $\epsilon$-$\delta$ definition of continuity is not to make people proficient in finding a $\delta$ for a particular situation like $f(x)=\sqrt{x}$. Rather, the purpose is to develop appreciation of the need for a formal definition of continuity. The examples where we have students find $\delta$ in specific situations are the necessary exercises that build this appreciation for the abstract definition.

These exercises of finding $\delta$ for simple, explicit functions are the scaffolding that enables a student to make sense of how continuity is invoked in the proof of the intermediate value theorem or the fundamental theorem of calculus, and to appreciate examples of functions that are continuous but nowhere differentiable, or discontinuous everywhere except at a point, for example.

If one's understanding of continuity is to ever develop beyond the "trace the curve without lifting the pencil" level, then exercises of finding $\delta$ in simple concrete situations are necessary. But proficiency in these exercises is not the goal. The goal is to provide scaffolding resulting in an understanding of the abstract definition, so one can make sense of continuity in theoretical situations.

Answer 2

I don't know whether this is a comment or an answer.

There is a bigger picture to keep in mind, which is that, beyond developing the students' understanding of the epsilon-delta definition and their ability to find delta, we also want to develop the students' cleverness. There's a certain art to finding the unnatural leaps that enable someone to make the delicate estimates that are necessary in more advanced (and difficult) pieces of analysis. Students need to learn that math (and life!) require cleverness and have opportunities to develop it.

Of course there is a need to balance competing goals.

Answer 3

But this leap (replacing $\sqrt{x}+2$ with $2$) seems so unnatural to students who are encountering this for the first time.

Students will encounter it eventually, for example in their next calculus course someone will ask them

Does this series converge? $\sum_{n=1}^\infty \frac{1}{\sqrt{n} + 2}$

and a solution might start with the fact that

For $n \geq 4, \frac{1}{\sqrt{n}+2} \geq \frac{1}{\sqrt{n} + \sqrt{n}} = \frac12 \frac{1}{\sqrt{n}}$.

So you might as well show them how to estimate expressions with other expressions now. It's just a useful tool.