AP Calculus AB Unit 1: Limits & Continuity
Study limit definition, limit properties, squeeze theorem, continuity with exam-format practice and rubric-based scoring.
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Inside This Unit: The Full Breakdown
Limits describe the behavior of a function as its input approaches a particular value, even when the function itself is undefined there. Continuity means a function has no breaks, jumps, or holes at a point.
Why it matters
Limits are the foundation of all calculus. Derivatives and integrals are both defined using limits, so mastering this unit is essential before anything else in the course. The AP exam frequently tests limit evaluation and continuity analysis in both multiple choice and free response.
Key concepts
- A limit describes what value f(x) approaches as x approaches c, which may differ from f(c) itself.
- Continuity at a point requires that the limit exists, the function is defined, and they are equal.
- The Squeeze Theorem lets you evaluate limits of complicated functions by bounding them between simpler ones.
- The Intermediate Value Theorem guarantees a continuous function takes every value between f(a) and f(b).
Understanding Limits Graphically and Numerically
A limit asks: what value does f(x) get close to as x gets close to some number c? You can estimate limits from a table by choosing x-values increasingly close to c from both sides. Graphically, you trace the curve toward x = c and observe the y-value it approaches. One-sided limits examine behavior from only the left or right. If the left-hand and right-hand limits disagree, the two-sided limit does not exist.
Algebraic Limit Techniques
Direct substitution is always the first strategy: plug in and see if you get a real number. When substitution yields 0/0, that signals an indeterminate form requiring algebraic manipulation. Factor and cancel, multiply by conjugates, or simplify complex fractions to resolve the indeterminate form. For limits at infinity, divide every term by the highest power of x in the denominator to determine horizontal asymptote behavior. These algebraic skills appear constantly on the AP exam.
Continuity and the Intermediate Value Theorem
A function is continuous at x = c when three conditions hold: f(c) is defined, the limit as x approaches c exists, and the limit equals f(c). Discontinuities come in three flavors: removable (holes), jump, and infinite (vertical asymptotes). The Intermediate Value Theorem says that if f is continuous on [a, b] and N is between f(a) and f(b), then f(c) = N for some c in (a, b). The IVT is commonly used on the AP exam to justify that a solution exists within an interval.
AP exam tip
When asked to justify that a value exists in an interval, cite the IVT by name, verify continuity on the closed interval, and show that the target value lies between f(a) and f(b).
Connections to other units
- Unit 2: The derivative is defined as a limit of a difference quotient, so limit skills transfer directly.
- Unit 6: The definite integral is defined as a limit of Riemann sums.
- Unit 5: Continuity conditions connect to analyzing function behavior and the Extreme Value Theorem.
