Limits and Continuity Overview — AP Calculus BC
1. Unit at a Glance
We open with the core question that motivates all of calculus: can change occur at an instant? This leads us to intuitively and formally define limits, then practice estimating limits from both graphs and tables.
After building intuition, we learn algebraic techniques to compute exact limits, then connect limits to the formal definition of continuity. We classify discontinuities, explore key theorems like the Intermediate Value Theorem, and finally connect limits to the asymptotic behavior of functions, tying together graphical and analytical representations.
Common Pitfalls
Why: The limit describes the behavior of $f(x)$ near $a$, not at $a$, so the two values can be entirely different
Why: Only point (removable) discontinuities can be removed; jump and infinite discontinuities cannot
Why: The IVT only holds for continuous functions on closed intervals, so it cannot be used for discontinuous functions