AP Calculus BC 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
All Questions (9)
A) 3
B) -10
C) 7
D) The limit cannot be determined.
Correct Answer: A
According to the limit theorem for sums, the limit of a sum of two functions is the sum of their individual limits. Therefore, lim(x→c) [f(x) + g(x)] = lim(x→c) f(x) + lim(x→c) g(x) = 5 + (-2) = 3.
A) 11
B) 18
C) 36
D) 9
Correct Answer: C
Using the limit theorems for a constant multiple and a product, we have: lim(x→-1) [2 * f(x) * g(x)] = 2 * [lim(x→-1) f(x)] * [lim(x→-1) g(x)] = 2 * 6 * 3 = 36.
A) 48
B) 3
C) 8
D) The limit does not exist.
Correct Answer: B
The limit theorem for quotients states that the limit of a quotient is the quotient of the limits, provided the limit of the denominator is not zero. Here, lim(x→8) [p(x) / q(x)] = [lim(x→8) p(x)] / [lim(x→8) q(x)] = 12 / 4 = 3.
A) 2
B) 9
C) 5
D) The limit cannot be determined.
Correct Answer: C
According to the limit theorem for composite functions, if f is continuous at L and lim(x→c) g(x) = L, then lim(x→c) f(g(x)) = f(L). In this case, c=2 and L=9. Therefore, lim(x→2) f(g(x)) = f(lim(x→2) g(x)) = f(9) = 5.
A) 6
B) -6
C) 0
D) The limit does not exist.
Correct Answer: B
To determine this one-sided limit analytically, we consider values of x approaching 3 from the left (x < 3). For x < 3, the term (x - 3) is negative, so |x - 3| = -(x - 3). The expression becomes (x - 3)(x + 3) / -(x - 3). Canceling the (x - 3) terms leaves -(x + 3). The limit is then lim(x→3⁻) -(x + 3) = -(3 + 3) = -6.
A) -7
B) -17
C) 1
D) 7
Correct Answer: B
This problem requires using the limit theorems for a constant multiple and a difference. lim(x→a) [3f(x) - g(x)] = 3 * [lim(x→a) f(x)] - [lim(x→a) g(x)] = 3 * (-4) - 5 = -12 - 5 = -17.
A) h(5) = 2
B) h(x) is continuous at x = 5.
C) lim(x→5) h(x) = 2
D) The graph of h(x) has a vertical asymptote at x = 5.
Correct Answer: C
The two-sided limit of a function exists if and only if the left-sided and right-sided limits both exist and are equal. Since lim(x→5⁺) h(x) = 2 and lim(x→5⁻) h(x) = 2, we can conclude that lim(x→5) h(x) = 2. We cannot determine the value of h(5) or continuity from the limit alone.
A) -18/7
B) -6
C) 6
D) The limit cannot be determined.
Correct Answer: B
This problem requires combining multiple limit theorems. First, find the limit of the numerator using the product rule: lim(x→4) [f(x) * g(x)] = 9 * (-2) = -18. Next, find the limit of the denominator using the sum rule: lim(x→4) [g(x) + 5] = -2 + 5 = 3. Finally, use the quotient rule: lim(x→4) [Numerator / Denominator] = -18 / 3 = -6.
A) -17
B) -1
C) 15
D) -9
Correct Answer: B
The function f(x) = x³ + 2x² - 1 is a polynomial. The limit of a polynomial function can be found by direct substitution, which is a consequence of the limit theorems for sums, differences, and powers. Substituting x = -2 gives: (-2)³ + 2(-2)² - 1 = -8 + 2(4) - 1 = -8 + 8 - 1 = -1.