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