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AP Calculus AB 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

A function f has a discontinuity at x=c. If the limit as x approaches c from the left is 5 and the limit as x approaches c from the right is -1, what type of discontinuity does f have at x=c?

All Questions (7)

A function f has a discontinuity at x=c. If the limit as x approaches c from the left is 5 and the limit as x approaches c from the right is -1, what type of discontinuity does f have at x=c?

A) Removable discontinuity

B) Jump discontinuity

C) Discontinuity due to a vertical asymptote

D) The function is continuous at x=c

Correct Answer: B

A jump discontinuity occurs when the left-hand limit and the right-hand limit both exist as finite numbers but are not equal to each other. Here, the left-hand limit is 5 and the right-hand limit is -1, so it is a jump discontinuity.

A function g is discontinuous at x=a. Which of the following conditions, if true, would classify the discontinuity at x=a as removable?

A) The limit of g(x) as x approaches a is infinity.

B) The limit of g(x) as x approaches a from the left is different from the limit as x approaches a from the right.

C) The limit of g(x) as x approaches a exists, but g(a) is undefined.

D) The limit of g(x) as x approaches a does not exist for reasons other than a jump or vertical asymptote.

Correct Answer: C

A removable discontinuity occurs when the limit of the function exists at a point, but the function is not continuous at that point. This happens if either the function value is undefined or the function value is not equal to the limit. Option C perfectly describes one of these scenarios.

According to the formal definition of continuity, a function f is discontinuous at x=c if 'lim (x->c) f(x) ≠ f(c)'. Which of the following must be true for this specific justification of discontinuity?

A) f(c) is undefined.

B) The limit of f(x) as x approaches c does not exist.

C) Both the limit of f(x) as x approaches c and the value f(c) must exist.

D) The function must have a jump discontinuity at x=c.

Correct Answer: C

The statement 'lim (x->c) f(x) ≠ f(c)' implies that both sides of the inequality are defined, finite values. Therefore, the limit must exist and the function value f(c) must be defined for the comparison to be made. This scenario describes a removable discontinuity.

A function h has a discontinuity at x=4. If the limit of h(x) as x approaches 4 from the right is positive infinity, what type of discontinuity must h have at x=4?

A) Removable discontinuity

B) Jump discontinuity

C) Discontinuity due to a vertical asymptote

D) Not enough information to determine

Correct Answer: C

A discontinuity is classified as being due to a vertical asymptote if either the left-hand or right-hand limit (or both) at that point is infinite (positive or negative infinity). Since the right-hand limit is positive infinity, the function has a vertical asymptote at x=4.

To justify that a function f is continuous at a point x=a, a student must verify three conditions from the definition of continuity. Which of the following is NOT one of those three required conditions?

A) f(a) must be defined.

B) The limit of f(x) as x approaches a must exist.

C) The graph of f must be a smooth curve at x=a.

D) The limit of f(x) as x approaches a must equal f(a).

Correct Answer: C

The three conditions for continuity at a point x=a are: 1) f(a) is defined, 2) the limit of f(x) as x approaches a exists, and 3) the limit equals the function value. The concept of a 'smooth curve' relates to differentiability, which is a stronger condition than continuity. A function can be continuous at a point without being smooth (e.g., a sharp corner).

If the limit of a function f(x) as x approaches c exists, but the function is discontinuous at x=c, what must be the type of discontinuity?

A) Jump discontinuity

B) Discontinuity due to a vertical asymptote

C) Removable discontinuity

D) It could be either a jump or a removable discontinuity.

Correct Answer: C

By definition, if the limit of f(x) exists at a point, it cannot be a jump discontinuity (where left and right limits differ) or a discontinuity due to a vertical asymptote (where the limit is infinite). The only type of discontinuity where the limit exists is a removable discontinuity, which occurs because f(c) is either undefined or not equal to the limit.

Consider a function g where g(2) = 5, the limit of g(x) as x approaches 2 from the left is 3, and the limit of g(x) as x approaches 2 from the right is 3. Which statement correctly justifies the continuity or discontinuity of g at x=2?

A) g is continuous at x=2 because g(2) is defined.

B) g has a jump discontinuity at x=2 because the limit and the function value are different.

C) g has a removable discontinuity at x=2 because the limit as x approaches 2 exists but is not equal to g(2).

D) g is continuous at x=2 because the left-hand and right-hand limits are equal.

Correct Answer: C

For continuity, all three conditions must be met. Here, g(2) is defined (it's 5). The limit as x approaches 2 exists because the left-hand limit (3) equals the right-hand limit (3), so the two-sided limit is 3. However, the third condition fails because the limit (3) does not equal the function value (5). This specific failure defines a removable discontinuity.