Experimentation, Part 2

I have another experiment with my calculus courses this fall. I've never been happy with continuity following direct substitution. It feels too much like being asked if your feet will get wet after you just put on wellies. The only functions you can (immediately) use direct substitution on are continuous functions! 

I want my limits and continuity unit to grow out of a review unit on functions and graphs. I want my students to look again at polynomials, rational functions, holes and asymptotes, and start putting together some observations about when there's a hole (or an asymptote) and when there isn't. I want them to talk about when we might "assume" a function to have a certain value, and when it would be wrong to have that "assumption". (We all know what happens when you assume... well, at least part of the time.)

At some point we'll switch our language from "assumption" to "limit". We'll make tables and look at graphs, and compare the actual function values to the limit values. And then we start categorizing, conjecturing and discussing, and viola! Continuity!

And from continuity? From that comes "Mrs. Hamilton, if we know this function is continuous (see? Look at the graph!), can't we just say that the limit of f(x) as x approaches a is whatever f(a) is?"

I can dream, right?

Now I just have to figure out how to work/weave in the properties of limits and continuity and more formal proof work.