Family Overview

Calculus BC · AP · Limits and Continuity

What this topic is

This topic opens with a guiding question: Introducing Calculus: Can Change Occur at an Instant? In Calculus BC · AP, students learn to answer it with clear vocabulary, representations, and justification—not only a quick numerical guess.

This topic is assessed on the AP Exam.

Why it matters

• This topic trains the precise “approaches” language AP graders look for on limit items.
• Limits and continuity are the language of Calculus AB/BC Unit 1 and appear across the AP Exam.
• Students must estimate from graphs/tables and compute with algebra—both styles are tested.
• Continuity and limit theorems unlock later derivative and integral justifications.
• Saying “approaches” correctly is half the battle on free-response write-ups.

How you can support

• Ask whether left- and right-hand behavior agree before concluding a two-sided limit.
• Ask: “From the left, from the right, or overall—what does the question want?”
• Have them point to a graph feature (hole, jump, asymptote) and name the limit.
• Praise correct limit notation even when the numerical value still needs work.
• For table estimates, ask whether values are approaching the same number.
• Keep calculator use for checking, not for replacing algebraic limit laws.

What not to do

• Do not equate a limit with a function value when the graph has a hole or jump.
• Do not say a limit “equals” a function value when there is a hole or jump.
• Do not cancel factors without noting how that changes the graph (removable discontinuity).
• Do not treat ∞ as a number you can plug in casually.
• Do not skip one-sided limits when the problem shows different left/right behavior.