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AP Calculus BC Practice Quiz: Exploring Types of Discontinuities

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 7 questions to check your progress.

Question 1 of 7

Let f be the function defined by f(x) = (x^2 - 9) / (x - 3). Which of the following statements describes the discontinuity of f at x = 3?

All Questions (7)

Let f be the function defined by f(x) = (x^2 - 9) / (x - 3). Which of the following statements describes the discontinuity of f at x = 3?

A) The function has a jump discontinuity because the one-sided limits are not equal.

B) The function has a removable discontinuity because the limit exists but the function is not defined at x = 3.

C) The function has a discontinuity due to a vertical asymptote because the limit approaches infinity.

D) The function is continuous at x = 3.

Correct Answer: B

The function can be simplified to f(x) = x + 3 for x ≠ 3. The limit as x approaches 3 is lim (x->3) (x + 3) = 6. However, f(3) is undefined. Since the limit exists but the function value does not, this is a removable discontinuity. The first condition for continuity (f(c) is defined) fails.

A function g(x) has a jump discontinuity at x = a. Which of the following conditions of the definition of continuity must have failed?

A) g(a) is not defined.

B) lim (x->a) g(x) does not exist.

C) lim (x->a) g(x) exists, but is not equal to g(a).

D) Both g(a) is undefined and lim (x->a) g(x) does not exist.

Correct Answer: B

A jump discontinuity is defined by the left-hand limit and the right-hand limit both existing as finite values, but not being equal to each other. If the one-sided limits are not equal, the overall limit, lim (x->a) g(x), does not exist. This violates the second condition for continuity.

Consider the function h(x) defined as: h(x) = { x + 2, if x < 1; 3, if x = 1; x^2 + 1, if x > 1 }. Which statement correctly justifies the type of discontinuity at x = 1?

A) h(x) has a jump discontinuity at x = 1 because lim (x->1-) h(x) = 3 and lim (x->1+) h(x) = 2.

B) h(x) has a removable discontinuity at x = 1 because lim (x->1) h(x) exists but is not equal to h(1).

C) h(x) has a removable discontinuity at x = 1 because h(1) is defined but the limit does not exist.

D) h(x) has a discontinuity due to a vertical asymptote at x = 1.

Correct Answer: A

To analyze the continuity at x = 1, we check the one-sided limits. The limit from the left is lim (x->1-) (x + 2) = 3. The limit from the right is lim (x->1+) (x^2 + 1) = 2. Since the left-hand limit (3) and the right-hand limit (2) are both finite but not equal, the function has a jump discontinuity at x = 1.

For a function f(x) at a point x = c, it is known that lim (x->c) f(x) exists and is finite, but the function is not continuous at x = c. This describes which type of discontinuity?

A) A jump discontinuity.

B) A removable discontinuity.

C) A discontinuity due to a vertical asymptote.

D) This situation is not possible for a discontinuity.

Correct Answer: B

The definition of continuity requires three things: f(c) is defined, the limit as x approaches c exists, and the limit equals the function value. If the limit exists but the function is not continuous, it must be because either f(c) is not defined or f(c) is not equal to the limit. Both of these cases define a removable discontinuity.

The function k(x) = 5 / (x - 4) has what type of discontinuity at x = 4?

A) Removable discontinuity

B) Jump discontinuity

C) Discontinuity due to a vertical asymptote

D) The function is continuous at x = 4

Correct Answer: C

As x approaches 4, the denominator (x - 4) approaches 0. This causes the value of the function k(x) to approach positive or negative infinity. This behavior indicates a vertical asymptote at x = 4, which is a type of non-removable discontinuity.

Let f be the function defined by f(x) = { 3x - 1, if x < 2; 5, if x > 2 }. Which of the following provides a complete justification regarding the continuity of f at x = 2?

A) The function is continuous because the left and right-hand limits are equal.

B) The function has a jump discontinuity because f(2) is undefined.

C) The function has a removable discontinuity because lim (x->2) f(x) = 5 but f(2) is undefined.

D) The function has a jump discontinuity because lim (x->2-) f(x) = 5 and lim (x->2+) f(x) = 5, but f(2) is undefined.

Correct Answer: C

First, we evaluate the one-sided limits. lim (x->2-) f(x) = 3(2) - 1 = 5. lim (x->2+) f(x) = 5. Since the one-sided limits are equal, the overall limit lim (x->2) f(x) exists and is equal to 5. However, the function is not defined at x = 2. According to the definition of continuity, since the limit exists but f(2) is not defined, the function has a removable discontinuity at x = 2.

If a function f has a discontinuity at x = c due to a vertical asymptote, which condition from the definition of continuity must fail?

A) Only the condition that f(c) must be defined.

B) Only the condition that lim (x->c) f(x) must equal f(c).

C) The condition that lim (x->c) f(x) must exist as a finite number.

D) All three conditions for continuity must fail.

Correct Answer: C

For a discontinuity due to a vertical asymptote, the limit of the function as x approaches c is either positive or negative infinity. Since the limit is not a finite number, the condition that lim (x->c) f(x) must exist (as a finite value) fails. This is the primary failure. Often, f(c) is also undefined, but the non-existence of a finite limit is the defining characteristic of this type of discontinuity.