Limits and Continuity Worksheet with Answers – AP Calculus AB (Free PDF)

Problem 1 Limits from a graph ★★★

The graph of h is shown. Find each value:

1. lim_x→1⁻ h(x)
2. lim_x→1⁺ h(x)
3. lim_x→1 h(x)
4. h(1)
5. lim_x→3 h(x)

Problem 2 Estimating from a table ★★★

A table of values for f(x) = x³ − 8x − 2 is given. Estimate lim_x→2 f(x).

Problem 3 Direct substitution ★★★

Evaluate lim_x→−2 (3x² − 4x + 1).

Problem 4 Factoring a 0/0 form ★★★

Evaluate lim_x→4 x² − x − 12x − 4.

Problem 5 Using a conjugate ★★★

Evaluate lim_x→0 √(x + 25) − 5x.

Problem 6 One-sided limits with absolute value ★★★

Let g(x) = |2x − 10|x − 5. Find lim_x→5⁻ g(x), lim_x→5⁺ g(x) and lim_x→5 g(x).

Problem 7 Limit at infinity of a rational function ★★★

Evaluate lim_x→∞ 6x² − x2x² + 5 and state the horizontal asymptote.

Problem 8 Limits at infinity with a root ★★★

Evaluate lim_x→∞ 4x + 1√(x² + 9) and lim_x→−∞ 4x + 1√(x² + 9).

Problem 9 Infinite one-sided limits ★★★

Let f(x) = x + 1(x − 7)(x + 3). Find lim_x→7⁺ f(x), lim_x→7⁻ f(x), and name every vertical asymptote.

Problem 10 Squeeze Theorem ★★★

Use the Squeeze Theorem to evaluate lim_x→0 x²cos(7/x).

Problem 11 Making a function continuous ★★★

Let f(x) = x² − 2x − 35x − 7 for x ≠ 7 and f(7) = k. Find k so that f is continuous at x = 7.

Problem 12 Classifying discontinuities ★★★

Let f(x) = (x + 5)(x − 6)(x + 5)(x − 2). List the discontinuities, state their types and give the value or behavior of f near each.

Problem 13 A piecewise function and one parameter ★★★

Let f(x) = x² + ax for x ≤ 2 and f(x) = 7x − 9 for x > 2. Find a so that f is continuous at x = 2.

Problem 14 Intermediate Value Theorem for a polynomial ★★★

Show that f(x) = x³ + 3x − 11 has a zero in the interval (1, 2).

Problem 15 Counting guaranteed zeros ★★★

The function g is continuous on [0, 6] and its known values are plotted.

1. What is the least number of zeros of g in (0, 6)?
2. Is g(x) = 4.5 guaranteed to have a solution in [4, 6]? Is it guaranteed in [0, 2]?

Problem 16 Parking garage fees ★★★

A garage charges $6 for a stay of up to 2 hours, $9 for more than 2 and up to 4 hours, and $12 for more than 4 and up to 6 hours. Let C(t) be the cost for a stay of t hours.

1. Find lim_t→4⁻ C(t) and lim_t→4⁺ C(t).
2. Is C continuous at t = 3? At t = 4?
3. What does a 4.5-hour stay cost?

Problem 17 Medicine level in the blood ★★★

The concentration of a medicine in a patient, in mg/L, is C(t) = 45t + 203t + 10 where t is hours after the dose. Find C(0), the limit as t → ∞, and the time when the concentration reaches 14 mg/L.

Problem 18 Repairing a cubic quotient ★★★

Let f(x) = x³ − 27x² − 9. Find every discontinuity, classify it, and tell how to redefine f to remove the removable one.

Problem 19 Horizontal and vertical asymptotes with a root ★★★

Let f(x) = √(9x² + 4x)x + 1. Find its horizontal asymptotes and describe its behavior at x = −1.

Problem 20 Continuity of a three-piece function ★★★

Let f(x) = 2x + a for x < −1; bx + 3 for −1 ≤ x ≤ 2; and x² + 1 for x > 2. Find a and b so that f is continuous for all real x.