Calculus courses usually start with limits, and the precise definition of a limit is usually included but barely. There are a number of problems with this.
First off, limits seem alien and "why are we doing this"? It would be better to start with the derivative, connect it to slopes and velocities, and tell a bit of the history (Newton attempting to understand gravity). Perhaps do some hand-waving about the limits, and promise to come back to them.
The second big problem is that the limit definition is never connected with the derivative definition. In fact, the f(x) in the limit definition is not the same function as the f(x) in the derivative definition. After almost 30 years of teaching calculus, I finally wrote something to explain how the two definitions interact. (I found nothing online about this, and I have never seen it in a textbook.)
I've written it up two ways. One is a bonus chapter for my (not-yet-published) book, Althea and the Mysteries of Calculus, in which there is discussion among the kids and 'Mom'. The other leaves out the characters: How the Definitions of Limit and Derivative Interact
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