AP Calculus AB Practice Quiz: Connecting Infinite Limits and Vertical Asymptotes
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
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Question 1 of 9
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A) As x approaches 2 from the right, the values of f(x) increase without bound.
B) As x approaches 2 from the left, the values of f(x) increase without bound.
C) The value of f(2) is infinity.
D) The graph has a horizontal asymptote at y=2.
Correct Answer: A
The notation x→2⁺ means x is approaching 2 from values greater than 2 (from the right). The limit evaluating to ∞ means the function's values, f(x), are increasing without bound. This describes unbounded behavior, which is characteristic of a vertical asymptote.
A) The graph has a hole at x = a.
B) The graph has a horizontal asymptote at y = a.
C) The graph has a vertical asymptote at x = a.
D) The function g(x) is defined at x = a.
Correct Answer: C
An infinite limit as x approaches a finite number 'a' is the definition of a vertical asymptote at x = a. The function's values exhibit unbounded behavior (in this case, decreasing without bound) as x gets arbitrarily close to 'a'.
A) The graph of h(x) has a vertical asymptote at x = -1, where the function increases without bound on both sides of the asymptote.
B) The graph of h(x) has a vertical asymptote at x = -1, where the function decreases without bound on both sides of the asymptote.
C) The graph of h(x) has a vertical asymptote at x = -1, where the function increases on the left and decreases on the right.
D) The graph of h(x) has a horizontal asymptote at y = -1.
Correct Answer: C
The limit from the left, lim_{x→-1⁻} h(x) = ∞, indicates that the function increases without bound as x approaches -1 from the left side. The limit from the right, lim_{x→-1⁺} h(x) = -∞, indicates that the function decreases without bound as x approaches -1 from the right side. This describes a vertical asymptote at x = -1 with opposing behaviors on each side.
A) The limit exists and is equal to a very large number.
B) The limit does not exist because the function's values are not approaching a specific real number.
C) The limit exists, and the function is defined as f(c) = ∞.
D) The limit exists only if the one-sided limits from the left and right are both equal to ∞.
Correct Answer: B
For a limit to exist in the formal sense, the function must approach a finite, real number L. An infinite limit is a specific way of describing why a limit does not exist. It describes unbounded behavior where the function's values grow infinitely large rather than converging to a single number.
A) lim_{x→4} f(x) = 0
B) lim_{x→∞} f(x) = 4
C) lim_{x→4} f(x) = ∞
D) lim_{x→0} f(x) = 1/16
Correct Answer: C
The function has a potential vertical asymptote where the denominator is zero, which is at x=4. As x approaches 4 from either the left or the right, the denominator (x-4)² approaches 0 through positive values. A constant numerator (1) divided by a quantity approaching 0 results in an unbounded function. Since the denominator is always positive, the function increases without bound towards ∞.
A) lim_{x→3⁻} g(x) = ∞ and lim_{x→3⁺} g(x) = ∞
B) lim_{x→3⁻} g(x) = -∞ and lim_{x→3⁺} g(x) = -∞
C) lim_{x→3⁻} g(x) = ∞ and lim_{x→3⁺} g(x) = -∞
D) lim_{x→-∞} g(x) = 3 and lim_{x→∞} g(x) = 3
Correct Answer: B
The phrase 'decreases without bound' is represented mathematically by a limit equaling -∞. The description states this happens as x approaches 3 from the left (x→3⁻) and also as x approaches 3 from the right (x→3⁺). Therefore, both one-sided limits must be equal to -∞.
A) lim_{x→2} f(x) = 1/4
B) lim_{x→-2⁻} f(x) = -∞
C) lim_{x→-2⁺} f(x) = -∞
D) lim_{x→-2} f(x) does not exist because the function oscillates.
Correct Answer: B
First, simplify the function: f(x) = (x-2) / ((x-2)(x+2)) = 1 / (x+2) for x ≠ 2. This reveals a vertical asymptote at x=-2 and a removable discontinuity (hole) at x=2. To analyze the asymptotic behavior, we check the one-sided limit as x approaches -2 from the left (x→-2⁻). For values of x slightly less than -2 (e.g., -2.01), the denominator x+2 is a small negative number. Therefore, the function 1/(x+2) approaches -∞. Option A describes the hole, not the asymptote.
A) f(k) is undefined.
B) The line x=k is a vertical asymptote for the graph of f(x).
C) lim_{x→k⁻} f(x) = lim_{x→k⁺} f(x)
D) The function f(x) must be a rational function.
Correct Answer: B
The definition of a vertical asymptote at x=k is that the limit of the function as x approaches k (from the left, right, or both) is either ∞ or -∞. Therefore, the given limit statement directly implies that x=k is a vertical asymptote. The value f(k) can be defined or undefined, as the limit is concerned with the behavior near k, not at k. The function need not be rational (e.g., f(x) = ln|x-k|).
A) A vertical asymptote at x=1 and a horizontal asymptote at y=3.
B) A vertical asymptote at x=3 and a horizontal asymptote at y=1.
C) A removable discontinuity at x=1 and a horizontal asymptote at y=3.
D) A vertical asymptote at x=1 and no horizontal asymptote.
Correct Answer: A
The limits lim_{x→1⁺} f(x) = -∞ and lim_{x→1⁻} f(x) = ∞ describe unbounded behavior as x approaches the finite value of 1. This is the definition of a vertical asymptote at the line x=1. The limit lim_{x→∞} f(x) = 3 describes the end behavior of the function; as x increases without bound, the function values approach 3. This is the definition of a horizontal asymptote at the line y=3.