On Thu, 28 Aug 2014 12:12:15 -0600, Wayne Bishop <email@example.com> wrote:
> My experience was very similar to Lou's but I do not consider that first > shallow understanding of limit to have been a waste of time anymore than > the first introduction to Mathematical Induction. Yes, most of those of > us who can go through the mechanics appropriately sort of have the > feeling that we are only parroting what our instructor wants. However, > over time, we cultivate a deeper understanding that I do not believe > occurs as easily without those first baby steps. >
I certainly had no intent to suggest otherwise, though I could probably been more explicit about my beliefs.
I *was* asked to think about the epsilon-delta definitions for limit and for continuity in freshman calculus--but not very deeply. I was required to learn the definitions, and to be able to recite them on exams, but I was never asked to use one of them in a proof. And, although my calculus sequence required that I learn what were then the standard freshman calculus proofs, I was told that I would not be asked to do any epsilon-delta proofs.
If you want to see the course I took, find a copy of the two-volume textbook "Calculus with Analytic Geometry," by Melcher P. Fobes and Ruth Smyth,
It wasn't until the last semester of my four-semester sequence that I was asked to do anything with epsilons, and that was to show that some easy sequences converged to their limits. I got the full epsilon-delta story in my advanced calculus course, and by the end of that course, I was showing, for example, that certain sequences of functions converged uniformly on certain intervals.
I've never thought that memorizing the definitions during my freshman year contributed very much to later understanding, though. The "baby steps" I took that year toward understanding the limit/continuity concepts were simply intuitional and computational. And yet, as Wayne has suggested, they weren't wasted efforts by any means.
- --Lou Talman Department of Mathematical & Computer Sciences Metropolitan State University of Denver