AP Calculus BC Practice Test (2026)

What is the limit of the function f(x) = (x^2 - 9) / (x - 3) as x approaches 3?

Evaluate the limit: lim (x→0) [(sqrt(x+4) - 2) / x]

A biologist modeling the concentration C(t) of a slowly degrading pollutant in a closed pond (measured in parts per million, where t is days since an initial dump) proposes the long-term model C(t) = (3t^2 - 5t + 1) / (2t^2 + 7t - 4). Field measurements confirm that for t values beyond 30 days, the pollutant concentration stabilizes rather than growing without bound or decaying to zero. The biologist wants to determine the long-run equilibrium concentration, i.e., the limit of C(t) as t grows very large. She notes that both the numerator and denominator are quadratic polynomials in t and that for t above 1,000 the higher-degree terms (3t^2 on top, 2t^2 on bottom) dominate the linear and constant contributions.

Based on the biologist's model, what is the long-run equilibrium concentration lim (t→∞) C(t), in parts per million?

For a function g(x), if lim (x→a-) g(x) = L and lim (x→a+) g(x) = M, what must be true for lim (x→a) g(x) to exist?

A graph of a function f(x) is shown. It has a jump discontinuity at x=2, with lim (x→2-) f(x) = 1 and lim (x→2+) f(x) = 3. Also, f(2) = 1. At x=4, there is a hole, with lim (x→4) f(x) = 2 but f(4) is undefined. The function is continuous everywhere else.

Based on the description of the graph of f(x), at which of the following x-values is the function f(x) NOT continuous?

An engineer is studying the curvature of a small-deflection elastic beam. For a dimensionless load parameter x close to zero, she has derived a correction factor g(x) = (e^x − 1 − x) / x^2 that quantifies how far the true beam response departs from a simple linear approximation. Direct substitution of x = 0 into g(x) produces the indeterminate form 0/0 because both numerator and denominator vanish. The engineer wants the baseline correction value g(0) = lim (x→0) g(x), which represents the second-order bending sensitivity at rest. She recalls that L'Hôpital's Rule may be applied whenever numerator and denominator both tend to 0 and that repeated application is legitimate as long as each new quotient remains indeterminate.

Using the engineer's model, what is lim (x→0) g(x)?

If 1 - x^2/4 <= f(x) <= 1 + x^2/2 for all x near 0, what is lim (x→0) f(x)?

Which of the following conditions is NOT required for a function f(x) to be continuous at x = c?

Consider the sequence defined by a_n = (n^2 + 1) / (2n^2 - 3n + 5).

What is the limit of the sequence a_n as n approaches infinity?

A continuous function f is defined on the closed interval [1, 5] such that f(1) = 10 and f(5) = 2. Which of the following statements must be true?

Evaluate the limit: lim (x→0) [x / |x|]

Consider the piecewise function: f(x) = { 2x + 1 for x < 1 { x^2 + 2 for x ≥ 1

What is lim (x→1) f(x)?

For which of the following functions does lim (x→2) f(x) = infinity?

What is the limit of the sequence a_n = (n!) / (n^n) as n approaches infinity?

A graph of a function h(x) is shown. At x=3, there is a vertical asymptote. At x=5, there is a hole in the graph, but the function value h(5) is defined at a different y-value.

Based on the description of the graph of h(x), for which x-value does lim (x→c) h(x) NOT exist?

Which of the following describes a removable discontinuity?

A continuous function f has the following values: f(-2) = 5 f(0) = 1 f(2) = -3 f(4) = 7

On which of the following intervals is the Intermediate Value Theorem guaranteed to show that f(x) = 0 for some x?