AP Calculus AB Practice Test (2026)

What is the value of the limit: \(\lim_{x \to 3} (2x^2 - 5x + 1)\)?

Evaluate the limit: \(\lim_{x \to 2} \frac{x^2 - 4}{x - 2}\).

Find the value of \(\lim_{x \to 0} \frac{\sqrt{x + 9} - 3}{x}\).

Consider the piecewise function \(f(x) = \begin{cases} x^2 + 1 & \text{if } x < 2 \\ 3x - 1 & \text{if } x \ge 2 \end{cases}\). What is \(\lim_{x \to 2^-} f(x)\)?

An engineer at a municipal water-treatment plant is modeling the fill rate (in liters per second) of a chemical reservoir during the first minute after the intake valve opens. For the first second (\(t < 1\), in seconds), the inflow follows a smooth ramp-up and is modeled by \(R(t) = t^2 + k\), where \(k\) is a calibration constant determined by the pre-existing residual flow. Starting at \(t = 1\) second, the inflow switches to the linear cruise model \(R(t) = 2t + 3\). The engineer needs a single continuous rate model with no sudden jump in flow at the one-second transition, so the ramp-up and cruise expressions must agree at \(t = 1\).

For the piecewise model \(R(t) = \begin{cases} t^2 + k & \text{if } t < 1 \\ 2t + 3 & \text{if } t \ge 1 \end{cases}\) described in the passage, what value of \(k\) makes \(R(t)\) continuous at \(t = 1\)?

If \(4x - 9 \le f(x) \le x^2 - 4x + 7\) for \(x \ge 0\), what is \(\lim_{x \to 4} f(x)\)?

A graph of a function \(g(x)\) is shown. The graph has a hole at \((2, 1)\), a vertical asymptote at \(x = 0\), and a horizontal asymptote at \(y = 2\). The function passes through \((1, 0)\) and \((3, 2)\). For \(x < 2\), the function approaches \(y=1\) as \(x \to 2\). For \(x > 2\), the function also approaches \(y=1\) as \(x \to 2\). At \(x=2\), the function is defined as \(g(2)=3\).

Based on the graph description of \(g(x)\), what is \(\lim_{x \to 2} g(x)\)?

A graph of a function \(h(x)\) is shown. The graph has a jump discontinuity at \(x=1\), where \(\lim_{x \to 1^-} h(x) = 2\) and \(\lim_{x \to 1^+} h(x) = 4\). The function is defined at \(h(1)=2\). At \(x=3\), there is a removable discontinuity (hole) where \(\lim_{x \to 3} h(x) = 5\) but \(h(3)\) is undefined. The function is continuous everywhere else.

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

A biologist studying a closed-system algal tank models the concentration of a dissolved nutrient, in milligrams per liter, at time \(x\) (in hours since inoculation) with the function \(C(x) = \frac{3x^2 - 2x + 1}{5x^2 + 4x - 7}\). The numerator represents total nutrient mass (in mg) delivered after subtracting uptake, and the denominator represents the effective mixing volume (in liters) that grows with time as circulation stabilizes. To calibrate a long-run disposal schedule, the biologist needs the steady-state concentration — the value the function approaches as the tank runs indefinitely (\(x \to \infty\)) — rather than an early-time reading.

Based on the passage, what is the steady-state concentration \(\lim_{x \to \infty} C(x)\) in the biologist's tank?

Let \(f(x)\) be a continuous function on the closed interval \([0, 5]\). If \(f(0) = 2\) and \(f(5) = 10\), which of the following statements must be true by the Intermediate Value Theorem (IVT)?

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

Consider the following table of values for a function \(f(x)\):\n\n| x | f(x) |\n|-------|---------|\n| 1.9 | 4.9 |\n| 1.99 | 4.99 |\n| 1.999 | 4.999 |\n| 2.0 | 3.0 |\n| 2.001 | 5.001 |\n| 2.01 | 5.01 |\n| 2.1 | 5.1 |

Based on the table, what is the best estimate for \(\lim_{x \to 2} f(x)\)?

For the function \(f(x) = \frac{x + 1}{x^2 - 4}\), which of the following describes the behavior of \(f(x)\) as \(x \to 2^+\)?

Evaluate \(\lim_{x \to 0} \frac{\sin(3x)}{x}\).

A graph of a function \(f(x)\) is provided. The graph shows a function that approaches \(y=2\) as \(x \to -\infty\) and \(y=2\) as \(x \to \infty\). At \(x=-1\), there is a vertical asymptote. At \(x=1\), there is a hole at \((1, 3)\) and \(f(1)=1\). At \(x=3\), the function is continuous and \(f(3)=4\).

Based on the graph description of \(f(x)\), which of the following statements is true?

Which of the following functions has a removable discontinuity at \(x = 2\)?

Consider the following table of values for a function \(g(x)\):\n\n| x | g(x) |\n|-------|---------|\n| 4.9 | 7.1 |\n| 4.99 | 7.01 |\n| 4.999 | 7.001 |\n| 5.0 | 2.0 |\n| 5.001 | 6.999 |\n| 5.01 | 6.9 |\n| 5.1 | 6.1 |

Based on the table, what is the best estimate for \(\lim_{x \to 5^-} g(x)\)?