Limits and Continuity: AP Calculus AB Math Test – PDF Download

AP Calculus Limits Test: questions 1–5

Question 1: Limits from a graph (2 points)

The graph of \(g\) is shown.

1. Find \(\lim_{x\to 1^{-}} g(x)\), \(\lim_{x\to 1^{+}} g(x)\), and \(\lim_{x\to 1} g(x)\) (write DNE if it does not exist). (1 point)
2. Find \(g(1)\) and \(\lim_{x\to 3} g(x)\). (1 point)

Question 2: Algebraic limits (2 points)

Evaluate each limit.

1. \(\lim_{x\to 4} \frac{x^{2}-16}{x-4}\) (1 point)
2. \(\lim_{x\to 0} \frac{\sqrt{x+9}-3}{x}\) (1 point)

Question 3: Infinity and asymptotes (2 points)

1. Evaluate \(\lim_{x\to \infty} \frac{6x^{3}-2x}{2x^{3}+5}\). (1 point)
2. Let \(f(x)=\frac{x+1}{x-2}\). Find \(\lim_{x\to 2^{+}} f(x)\) and \(\lim_{x\to 2^{-}} f(x)\). (1 point)

Question 4: Continuity of a defined function (3 points)

Let \(f(x)=\frac{x^{2}-9}{x-3}\) for \(x\neq 3\) and \(f(3)=k\).

1. Find \(\lim_{x\to 3} f(x)\). (1 point)
2. Find the value of \(k\) that makes \(f\) continuous at \(x=3\). (1 point)
3. If \(k=2\), classify the discontinuity at \(x=3\). (1 point)

Question 5: Squeeze Theorem (2 points)

1. The function \(g\) satisfies \(4-(x-1)^{2}\le g(x)\le 4+(x-1)^{2}\) for all \(x\). Find \(\lim_{x\to 1} g(x)\) and justify. (1 point)
2. Evaluate \(\lim_{x\to 0} x^{2}\sin\frac{5}{x}\) and justify. (1 point)

AP Calculus Limits Test: questions 6–9

Question 6: Intermediate Value Theorem (3 points)

Let \(f(x)=x^{3}-x-3\). The graph is shown.

1. Use the Intermediate Value Theorem to show that \(f(x)=0\) has a solution in \((1,2)\). (2 points)
2. Use \(f(1.5)\) to narrow the interval that contains a solution. (1 point)

Question 7: Types of discontinuity (2 points)

The graph of \(q\) is shown.

1. Find every \(x\)-value where \(q\) is not continuous. (1 point)
2. Classify each discontinuity from part (a) and give its reason. (1 point)

Question 8: Piecewise continuity (2 points)

Let \(f(x)=kx+2\) when \(x\le 1\) and \(f(x)=3x^{2}-k\) when \(x > 1\). Find the value of \(k\) that makes \(f\) continuous at \(x=1\).

Question 9: Long-run concentration (2 points)

The concentration of a medicine in mg/L is \(C(t)=\frac{50t}{t+4}\), where \(t\) is hours after a dose.

1. Find \(\lim_{x\to \infty} C(t)\) and explain what it means. (1 point)
2. At what time does the concentration reach 40 mg/L? (1 point)

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