Unit 1 Practice Test: Limits and Continuity (AP Calculus BC)

Limits and Continuity in AP Calculus BC

Unit 1 establishes the analytical foundation for everything that follows in AP Calculus BC. Although this unit shares its content with AP Calculus AB, BC students encounter it at a faster pace — making it critical to achieve solid mastery before the course accelerates into differentiation, integration, and ultimately the BC-exclusive units on parametric functions and infinite series.

Core Topics Covered in Unit 1

Evaluating Limits

Students must be able to determine limits using multiple representations — graphs, tables of values, and algebraic expressions. This includes recognizing when direct substitution applies and when algebraic manipulation, such as factoring or rationalizing, is required.

One-Sided and Infinite Limits

One-sided limits (from the left and right) are tested both graphically and analytically. Infinite limits and limits at infinity relate directly to horizontal and vertical asymptote behavior, which appears throughout the BC exam.

Continuity and Types of Discontinuity

A function is continuous at a point when the limit exists, the function is defined, and both values agree. Students should be able to identify and classify discontinuities: removable (holes), jump, and infinite (asymptotic). These distinctions appear in both multiple-choice and free-response questions.

The Intermediate Value Theorem (IVT)

The IVT guarantees the existence of a value within an interval under specific conditions. AP-style questions often require students to cite and apply the IVT as a justification step in free-response problems — a skill that carries over into later units.

Why a Strong Limits Foundation Matters in BC

In AP Calculus BC, limits reappear throughout the course — from L'Hôpital's Rule in Unit 4 to the limit definition of derivatives, convergence of sequences in Unit 10, and improper integrals in Unit 6. A weak understanding of limits in Unit 1 creates compounding difficulty in every subsequent unit. BC students who treat limits as a warmup rather than a core skill frequently encounter avoidable errors in series convergence and polar function analysis later in the year.

What to Expect on Unit 1 Practice Tests

Unit 1 practice questions include limit evaluation from graphs and tables, algebraic limit problems requiring factoring and rationalization, continuity analysis at specific points, classification of discontinuity types, and IVT application and justification. Both multiple-choice and free-response style questions are included to reflect the full AP exam format.