AP Calculus AB Practice Quiz: Determining Limits Using Algebraic Properties 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 9 questions to check your progress.

Question 1 of 9

Given that lim(x→c) f(x) = L and lim(x→c) g(x) = M, which of the following represents the limit of the sum of the two functions as x approaches c?

A)L + MB)L - MC)L * MD)L / M

All Questions (9)

Given that lim(x→c) f(x) = L and lim(x→c) g(x) = M, which of the following represents the limit of the sum of the two functions as x approaches c?

A) L + M

B) L - M

C) L * M

D) L / M

Correct Answer: A

According to the limit theorems, the limit of a sum of two functions is equal to the sum of their individual limits. Therefore, lim(x→c) [f(x) + g(x)] = lim(x→c) f(x) + lim(x→c) g(x) = L + M.

If lim(x→5) f(x) = 10 and lim(x→5) g(x) = -2, what is the value of lim(x→5) [f(x) * g(x)]?

A) 8

B) -5

C) 12

D) -20

Correct Answer: D

The limit theorem for products states that the limit of the product of two functions is the product of their limits. Therefore, lim(x→5) [f(x) * g(x)] = (lim(x→5) f(x)) * (lim(x→5) g(x)) = 10 * (-2) = -20.

Let lim(x→a) f(x) = 12 and lim(x→a) g(x) = 4. What is lim(x→a) [f(x) / g(x)]?

A) 48

B) 3

C) 8

D) The limit cannot be determined.

Correct Answer: B

Based on the limit theorem for quotients, the limit of a quotient of two functions is the quotient of their limits, provided the limit of the denominator is not zero. Here, lim(x→a) [f(x) / g(x)] = (lim(x→a) f(x)) / (lim(x→a) g(x)) = 12 / 4 = 3.

According to the provided content, which of the following methods can be used to determine one-sided limits?

A) Analytically or graphically

B) Only analytically using limit theorems

C) Only by examining a table of values

D) Only graphically

Correct Answer: A

The content explicitly states that 'One-sided limits can be determined analytically or graphically.' [cite: 1233]. This means both algebraic methods (analytical) and visual inspection of a graph are valid approaches.

If lim(x→-1) f(x) = 6 and lim(x→-1) g(x) = 2, evaluate lim(x→-1) [3f(x) - g(x)].

A) 12

B) 16

C) 18

D) 4

Correct Answer: B

This problem uses the limit theorems for a constant multiple and a difference. lim(x→-1) [3f(x) - g(x)] = lim(x→-1) 3f(x) - lim(x→-1) g(x) = 3 * (lim(x→-1) f(x)) - lim(x→-1) g(x) = 3 * 6 - 2 = 18 - 2 = 16.

Let lim(x→4) g(x) = 9 and lim(u→9) f(u) = 5. What is the value of lim(x→4) f(g(x))?

A) 4

B) 9

C) 5

D) The limit cannot be determined from the information given.

Correct Answer: C

The limit theorem for composite functions states that lim(x→c) f(g(x)) = f(lim(x→c) g(x)). In this case, lim(x→4) f(g(x)) = f(lim(x→4) g(x)) = f(9). Since we are given that lim(u→9) f(u) = 5, the value is 5.

Given lim(x→c) f(x) = -3 and lim(x→c) g(x) = 5, find the value of lim(x→c) [f(x) + 2g(x)] / f(x).

A) 7/(-3)

B) 2

C) -7/3

D) 1

Correct Answer: C

This requires applying multiple limit theorems. First, evaluate the numerator: lim(x→c) [f(x) + 2g(x)] = lim(x→c)f(x) + 2*lim(x→c)g(x) = -3 + 2*(5) = -3 + 10 = 7. Then, apply the quotient rule: (lim(x→c) [f(x) + 2g(x)]) / (lim(x→c) f(x)) = 7 / (-3) = -7/3.

The limit theorem for quotients, lim(x→c) [f(x) / g(x)], can be applied directly by calculating the quotient of the individual limits only under which critical condition?

A) lim(x→c) f(x) ≠ 0

B) The function f(x) is continuous at x=c.

C) lim(x→c) g(x) ≠ 0

D) The functions f(x) and g(x) must be polynomials.

Correct Answer: C

The limit theorem for quotients states that the limit of a quotient is the quotient of the limits, but this is only valid if the limit of the function in the denominator is not zero. Division by zero is undefined, and this principle extends to the evaluation of limits.

Suppose f(x) = x³ and lim(x→-2) g(x) = 1. Using the limit theorems for composite functions, what is lim(x→-2) g(f(x))?

A) -8

B) 1

C) -6

D) The limit cannot be determined from the information given.

Correct Answer: D

The limit theorem for composite functions states lim(x→c) g(f(x)) = g(lim(x→c) f(x)). First, we find lim(x→-2) f(x) = lim(x→-2) x³ = (-2)³ = -8. The theorem then requires us to find g(-8). However, we are only given the value of lim(x→1) g(x), not lim(x→-8) g(x) or g(-8). Therefore, the limit cannot be determined.