A message last night from a friend pointed out that this chapter may contain too much and might be unfocused. It was a great suggestion. I had just sent a note off to my editor saying that I had decided that chapter two would be about limits and that continuity would now become chapter three.
The down side is that this seems to extend the discussion of these two topics while the student is raring to go and wants to get to the good stuff. As they will later learn, however, this is the good stuff. So many of the theorems require an understanding of these two topics. The one I was tempted to give short shrift to is continuity – we’ll see.
As for limits, I’m leaving the formal definition until the end of the chapter. This means that I’m not providing formal proofs of any of the limit theorems. That may be ok. I want students to understand that if the limit as x -> a f(x) = 7 then lim x->a [f(x) + 3] is 10. I’ve seen tons of students who know the limit of the sum is the sum of the limits when they are studying that section of the book – can we get them to grasp these theorems at a gut level.
It also sets up nicely the expectation that the derivative of the product is the product of the derivatives.