AP Calculus BC Unit 1: Limits & Continuity
Study limit definition, limit properties, squeeze theorem, types of discontinuities with exam-format practice and rubric-based scoring.
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Inside This Unit: The Full Breakdown
Limits describe how a function behaves near a point or as x grows without bound. Continuity ensures a function has no gaps, and together these concepts form the foundation for all of calculus.
Why it matters
AP Calculus BC covers limits at the same depth as AB, but BC students must move through this material efficiently because additional topics await. Strong limit skills are essential for the series convergence tests that are unique to BC.
Key concepts
- Limits can be evaluated graphically, numerically, and algebraically — direct substitution is always the first approach.
- Continuity at a point requires the function to be defined, the limit to exist, and both to be equal.
- The Squeeze Theorem and IVT are essential tools for justifying existence of limits and values.
- Limits at infinity determine horizontal asymptotes by comparing growth rates of numerator and denominator.
Limit Evaluation Strategies
Start with direct substitution. If the result is a real number, that is the limit. If you get 0/0, factor, multiply by conjugates, or simplify to eliminate the indeterminate form. For limits at infinity, divide by the highest power of x in the denominator. Compare degrees: if the numerator degree exceeds the denominator, the limit is infinite; if equal, it is the ratio of leading coefficients; if less, the limit is zero. These techniques extend to exponential and logarithmic comparisons in later units.
Types of Discontinuity
Removable discontinuities occur when the limit exists but either the function is undefined or its value differs from the limit. Jump discontinuities occur when left and right limits both exist but are unequal. Infinite discontinuities occur at vertical asymptotes where the function grows without bound. Piecewise functions are a common source of AP questions about continuity. Always check all three conditions of the continuity definition at the boundary between pieces.
Theorems Involving Continuity
The Intermediate Value Theorem guarantees that a continuous function on [a, b] takes every value between f(a) and f(b). The Extreme Value Theorem guarantees a continuous function on a closed interval achieves both an absolute max and an absolute min. Both theorems require continuity on a closed interval. The AP exam asks you to verify hypotheses and cite these theorems by name. In BC, these foundational ideas also support the convergence arguments you will use for infinite series.
AP exam tip
On BC, limit questions may involve parametric or polar expressions. Make sure you can evaluate limits of the form dy/dx = (dy/dt)/(dx/dt) as t approaches a value where both numerator and denominator approach zero.
Connections to other units
- Unit 2: The derivative is a limit of a difference quotient — this is the bridge from limits to calculus.
- Unit 10: Series convergence tests (ratio, comparison) are fundamentally limit computations.
- Unit 9: Parametric derivatives involve limits when dx/dt = 0.
