limits

[skolem_hull]

Annoyed about how the textbook from which I'm teaching calculus implicitly quantifies over the reals when it defines limits. Instead of quantifying over the domain. Also instead of being explicit about anything.

(So sqrt(x) only has a right-sided limit at 0, not a limit.)

It might be the right thing to do (I'm not sure), but it's still annoying.

Comments

[crystalpyramid]

What would you rather it do?

[skolem_hull]

I'd rather intersect balls (with the centers deleted) of radius \delta with the domain of the function.

So then the quantification is "whenever x is in the interval (c-\delta,c+\delta) and not equal to c, ..." or maybe "whenever 0 < |x-c| < \delta and x-c is in the domain

[skolem_hull]

To clarify (3): We want to be able to talk about a limit of, say, sqrt(x), at 0. But the text's definition doesn't work there, so one-sided limits are introduced. This ad hoc solution works because the domain on which sqrt(x) is defined is very simple, a half-line. But for more general domains (for for sets of real numbers), ad hoc solutions won't work.

One-sided limits really shouldn't (aren't) an important concept. But talking about the limit of sqrt(x) at 0 is important. So we introduce them.

With the better definition of limit (the one mathematicians use, I think), the concept of one-sided limits doesn't need to be introduced except for very specialized purposes (this topic in fairly advanced probability theory is the only time I've ever seen one-sided limits used for actually doing math).

[andrylisse]

My gut reaction to the one-sided limit thing is that there's some process calculus students often learn for finding limits that makes more sense (perhaps only to calculus students) for one-sided limits. But I can't justify this off the top of my head

[skolem_hull]

My list of reasons for liking the 'better' definition was (inappropriately) aimed at mathematics generally. For this class, I have to (whether it's right or not-- yeah, my department is silly) get them to understand some collection of concepts with formal definitions. Those definitions do serve your roles (2) and (3), and even (1). But yeah, of course a different emphasis might be better.

So I'm thinking about what definitions are best for those concepts. Tomorrow I'll define "continuous on the interval [a,b]" as continuous at each point in (a,b), right continuous at a, and left continuous at b. Maybe it's good to emphasize how what's going on at the endpoints is different from the rest of the interval. And yeah, probably a definition I like better would be even harder to get across. I'm just not sure. And there are benefits (in terms of the role you give calculus education, which does seem right to me) of having fewer definitions.