In an ideal world, I would like to have all students study limits in their pre-calculus course and know all about them when they get to calculus. Certainly, this would be better than teaching how to calculate derivatives in pre-calculus (after all derivatives are calculus, not pre-calculus).
Limits are the foundation of the calculus. Continuity, an important property of functions, depends on limits. All derivatives and all definite integrals are limits. For AP Calculus students need a good intuitive understanding of limits, what they mean, and how to find them The formal (delta-epsilon) definition is not tested and need not be taught, however, do not feel that you have to avoid it. If your students can handle it, let them try.
Here are a few of my previous posts on limits.
Finding Limits How to … and the use of “infinity” vs “DNE”
Dominance Finding limits the easy way.
Deltas and Epsilons Not tested on the AP Exams; here’s why.
Asymptotes The graphical manifestation of limits at or equal to infinity.
Next Week: Continuity
Update of my post of August 15, 2017.
Teaching Calculus 