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AP Calculus BC Practice Quiz: Connecting Multiple Representations of Limits

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

Which of the following graphs visually represents the limit statement lim (x→2) f(x) = 1?

All Questions (7)

Which of the following graphs visually represents the limit statement lim (x→2) f(x) = 1?

A) A graph where the function approaches a y-value of 1 from both the left and right side of x=2, with a hole at the point (2, 1).

B) A graph with a solid point at (2, 1) but approaching a y-value of 3 from both sides of x=2.

C) A graph with a vertical asymptote at x=2.

D) A graph that shows a jump discontinuity at x=2, approaching y=1 from the left and y=2 from the right.

Correct Answer: A

The analytic notation lim (x→2) f(x) = 1 indicates that as x gets closer to 2 from both sides, the value of the function f(x) gets closer to 1. This does not depend on the actual value of f(2), which could be undefined (a hole) or a different value. Option A correctly depicts this behavior graphically.

Consider the following table of values for a function f(x). | x | f(x) | |-------|--------| | 2.9 | -4.9 | | 2.99 | -4.99 | | 2.999 | -4.999 | | 3.001 | -5.001 | | 3.01 | -5.01 | | 3.1 | -5.1 | Which analytic expression correctly describes the limit suggested by this numerical data?

A) lim (x→3) f(x) = -5

B) lim (x→-5) f(x) = 3

C) lim (x→3) f(x) = ∞

D) lim (x→3) f(x) does not exist

Correct Answer: A

The table shows that as the x-values approach 3 from both the left (2.9, 2.99, 2.999) and the right (3.1, 3.01, 3.001), the corresponding f(x) values approach -5. This numerical representation corresponds to the analytic limit statement lim (x→3) f(x) = -5.

Based on the provided graph of the function g(x), what is the value of lim (x→-1⁺) g(x)?

A) -2

B) 1

C) 3

D) The limit does not exist.

Correct Answer: B

The notation lim (x→-1⁺) g(x) asks for the value the function approaches as x approaches -1 from the right side (i.e., with values greater than -1). Following the graph from the right towards x=-1, the y-value of the function approaches 1. Therefore, the right-hand limit is 1.

The statement lim (x→a) f(x) = L is an analytic representation of a limit. Which of the following statements provides a correct interpretation of this notation?

A) The value of the function at x=a must be equal to L.

B) The function f(x) must be continuous at x=a.

C) As x approaches a, the value of f(x) gets arbitrarily close to L.

D) The graph of f(x) must have a solid point at (a, L).

Correct Answer: C

The definition of a limit states that lim (x→a) f(x) = L means that the function's values, f(x), can be made arbitrarily close to L by taking x sufficiently close to a, but not equal to a. The limit is about the approaching behavior, not the actual value of the function at the point x=a. Therefore, options A, B, and D, which make claims about the function's value or continuity at x=a, are not necessarily true.

If it is known that lim (x→0) h(x) does not exist, which of the following representations could describe the behavior of the function h(x) near x=0?

A) A graph of h(x) that is a smooth, continuous curve passing through the origin (0,0).

B) A table of values where h(x) approaches 4 as x approaches 0 from both the left and the right.

C) The analytic definition h(x) = (sin(x))/x.

D) A graph of h(x) showing the function approaching y=2 as x approaches 0 from the left, and approaching y=-1 as x approaches 0 from the right.

Correct Answer: D

A limit fails to exist if the left-hand limit and the right-hand limit are not equal. Option D describes a jump discontinuity where lim (x→0⁻) h(x) = 2 and lim (x→0⁺) h(x) = -1. Since the one-sided limits are different, the overall limit does not exist. The other options all describe scenarios where the limit at x=0 does exist.

A student creates a table of values to investigate a limit numerically. | x | f(x) | |--------|---------| | -1.1 | -10 | | -1.01 | -100 | | -1.001 | -1000 | | -0.999 | 1000 | | -0.99 | 100 | | -0.9 | 10 | Which graphical representation is consistent with the numerical data in the table?

A) A graph with a hole at x = -1.

B) A graph with a vertical asymptote at x = -1.

C) A graph with a jump discontinuity at x = -1.

D) A graph that is continuous at x = -1.

Correct Answer: B

The table shows that as x approaches -1 from the left, f(x) decreases without bound (approaches -∞). As x approaches -1 from the right, f(x) increases without bound (approaches +∞). This behavior, where the function's magnitude grows infinitely large as x approaches a specific value, is represented graphically by a vertical asymptote.

The limit of a function f(x) is expressed analytically as lim (x→4) f(x) = 2. Which of the following representations is NOT consistent with this statement?

A) A graph of a function that approaches a y-value of 2 as x approaches 4, with a solid point at (4, 5).

B) A table of values for f(x) that get closer and closer to 2 as the x-values get closer and closer to 4.

C) A graph of a function with a vertical asymptote at x=4.

D) The analytic definition f(x) = 2 for all x ≠ 4, and f(4) is undefined.

Correct Answer: C

The statement lim (x→4) f(x) = 2 means the function approaches a finite value, 2, as x approaches 4. A vertical asymptote at x=4 (Option C) would mean the function approaches ±∞, so the limit would not be 2. Options A, B, and D all describe scenarios where the function values approach 2 as x approaches 4, which is consistent with the given analytic statement.