Limits and Continuity Test – AP Calculus AB Math (Printable PDF)

Question 1 Evaluate a limit (2 points)

Which value equals lim_x→2 x² − x − 2x² − 4?

1. 0
2. 3/4
3. 3/2
4. The limit does not exist

Question 2 Reading a graph (3 points)

The graph of k is shown. Find:

1. lim_x→2 k(x) (1 point)
2. lim_x→4⁻ k(x), lim_x→4⁺ k(x) and k(4) (1 point)
3. Is k continuous at x = 4? Justify. (1 point)

Question 3 Limit at infinity (2 points)

Evaluate lim_x→∞ 7x³ − x2 − 5x³.

1. −7/5
2. −7/2
3. 7/5
4. The limit is ∞

Question 4 Conjugate method (2 points)

Show your work to evaluate lim_x→0 √(x + 16) − 4x.

Question 5 Continuity at a point (3 points)

Let f(x) = 2x² − 11x − 6x − 6 for x ≠ 6, and f(6) = 10.

1. Find lim_x→6 f(x). (1 point)
2. Is f continuous at x = 6? Use the three-part definition. (1 point)
3. Find the value of f(6) that would make f continuous, and classify the original discontinuity. (1 point)

Question 6 Discontinuities of a rational function (3 points)

Let f(x) = x² − x − 6x² − 9.

1. Find all discontinuities and classify each. (2 points)
2. Find lim_x→−3⁻ f(x) and lim_x→−3⁺ f(x). (1 point)

Question 7 Intermediate Value Theorem (2 points)

Let f(x) = x⁴ + x − 9. Explain why f(x) = 0 has a solution in (1, 2). Name the theorem and check its hypotheses.

Question 8 Asymptotes and one-sided limits (3 points)

The graph of f(x) = x + 2(x − 5)(x + 3) is shown.

1. Find lim_x→5⁺ f(x) and lim_x→5⁻ f(x). (1 point)
2. Find lim_x→−3⁻ f(x) and lim_x→−3⁺ f(x). (1 point)
3. Find the horizontal asymptote and justify. (1 point)