AP Calculus AB and BC
Unit 1: Limits and Continuity
Exam weighting: AB 10-15% · BC 5-10%
Unit 1 builds the limit: the tool for describing how a function behaves near a point it may never reach, then uses it to define continuity. It is 10-15% of the AB multiple-choice section and 5-10% of BC across 16 topics. The tested skill is picking the limit method that fits the form you get after substituting.
A limit describes what value a function approaches as the input nears a point, whether or not the function is even defined there (Topic 1.2). That gap between where a function is headed and where it actually lands is what makes calculus possible: it lets you define an instantaneous rate of change as the limit of average rates, which otherwise divide by zero at a single instant (Topic 1.1). You will express limits four ways: graphically, numerically, analytically, and verbally, but the formal epsilon-delta definition is not assessed on the AB or BC exam.
| Topics | What they cover | Big idea |
|---|---|---|
| 1.1 | Why an average rate breaks at an instant, and how a limit fixes it | Change (CHA-1) |
| 1.2-1.9 | Defining, estimating, and evaluating limits, and selecting the procedure | Limits (LIM-1) |
| 1.10-1.15 | Continuity, types of discontinuity, and infinite or end behavior | Limits (LIM-2) |
| 1.16 | The Intermediate Value Theorem, the first existence theorem | Analysis of Functions (FUN-1) |
Topic 1.7 is the heart of the unit and the reason a solver cannot do it for you: given a limit, you have to recognize which technique applies. The move is always the same. Substitute first. If you get a real number, that is the answer. If you get , a common factor is hiding, so factor or multiply by a conjugate (Topic 1.6). A nonzero number over signals a vertical asymptote (Topic 1.14). A shape points to a known trig limit or the squeeze theorem (Topic 1.8). Reading that returned form, not memorizing five procedures, is what the exam rewards.
Continuity is where limits pay off. A function is continuous at when three things all hold: exists, exists, and the two are equal (Topic 1.11). When they fail, the break is one of three named types: removable, jump, or infinite (a vertical asymptote), and identifying which is a common exam task (Topic 1.10). Polynomial, rational, power, exponential, logarithmic, and trigonometric functions are continuous everywhere in their domains (Topic 1.12), which is exactly why direct substitution works for them.
Unit 1 is 10-15% of the AB multiple-choice section and 5-10% of BC, and it appears on both the calculator and no-calculator parts. Expect to read limits off graphs and tables (Topics 1.3, 1.4), justify continuity at the boundary of a piecewise function (Topic 1.13), and apply the Intermediate Value Theorem. That last task tests a habit you will use all year: state and check a theorem's hypotheses before using its conclusion. The IVT guarantees a solution only after you confirm the function is continuous on a closed interval (Topic 1.16). Everything downstream rests on this unit, because every derivative and integral is itself a limit.
Topics in this unit
Topic numbers and titles from the College Board Course and Exam Description.
- 1.1Introducing Calculus: Can Change Occur at an Instant?
- 1.2Defining Limits and Using Limit Notation
- 1.3Estimating Limit Values from Graphs
- 1.4Estimating Limit Values from Tables
- 1.5Determining Limits Using Algebraic Properties of Limits
- 1.6Determining Limits Using Algebraic Manipulation
- 1.7Selecting Procedures for Determining Limits
- 1.8Determining Limits Using the Squeeze Theorem
- 1.9Connecting Multiple Representations of Limits
- 1.10Exploring Types of Discontinuities
- 1.11Defining Continuity at a Point
- 1.12Confirming Continuity over an Interval
- 1.13Removing Discontinuities
- 1.14Connecting Infinite Limits and Vertical Asymptotes
- 1.15Connecting Limits at Infinity and Horizontal Asymptotes
- 1.16Working with the Intermediate Value Theorem (IVT)
How to study this unit
- Run one loop on every limit: substitute first (Topic 1.5), then let the result pick the method. A real number is the answer, $\frac{0}{0}$ means factor or use a conjugate (Topic 1.6), and a nonzero number over $0$ is a vertical asymptote (Topic 1.14). That recognition is the Topic 1.7 skill the exam actually grades.
- Memorize the two squeeze-theorem limits cold: $\lim_{x \to 0} \frac{\sin x}{x} = 1$ and $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$ (Topic 1.8). They show up disguised, like $\frac{\sin 3x}{4x}$, so drill rewriting a problem into the standard shape.
- Treat the continuity definition as a three-box checklist (Topic 1.11): $f(c)$ exists, the limit exists, and they match. Piecewise boundary problems (Topic 1.13) are graded on showing all three, not on the arithmetic alone.
- For any Intermediate Value Theorem question (Topic 1.16), write the hypotheses before the conclusion: state that the function is continuous on the closed interval and that your target value lies between the endpoint values. No credit follows until continuity is verified.
- Handle limits at infinity by comparing leading degrees (Topic 1.15) instead of plugging in a large number: the ratio of the leading terms gives the horizontal asymptote and the end behavior directly.