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AP Calculus AB Practice Quiz: Connecting Limits at Infinity and Horizontal Asymptotes

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 10 questions to check your progress.

Question 1 of 10

The mathematical concept of a limit at infinity is primarily used to describe which feature of a function's graph?

All Questions (10)

The mathematical concept of a limit at infinity is primarily used to describe which feature of a function's graph?

A) The function's value at x = 0

B) The locations of vertical asymptotes

C) The function's end behavior

D) The instantaneous rate of change at a point

Correct Answer: C

Limits at infinity, such as lim x→∞ f(x) or lim x→-∞ f(x), describe the long-term trend or end behavior of a function as the input variable x increases or decreases without bound. [cite: 1474]

If lim x→∞ f(x) = 7, which of the following statements provides the best interpretation of this limit?

A) The graph of f(x) has a vertical asymptote at x = 7.

B) The value of f(x) approaches 7 as x increases without bound.

C) The function f(x) is undefined at x = 7.

D) The y-intercept of the graph of f(x) is at (0, 7).

Correct Answer: B

The notation lim x→∞ f(x) = 7 means that as the input x gets arbitrarily large, the output values of the function f(x) get arbitrarily close to 7. This describes the behavior of the function approaching a horizontal asymptote at y = 7. [cite: 1440, 1467]

To compare the relative magnitudes of the functions f(x) = eˣ and g(x) = x⁵⁰ as x gets very large, which of the following limits would be most useful?

A) lim x→0 (f(x) / g(x))

B) lim x→∞ (f(x) - g(x))

C) lim x→∞ (f(x) / g(x))

D) lim x→50 (f(x) / g(x))

Correct Answer: C

The relative magnitudes and rates of change of two functions as x approaches infinity are typically compared by taking the limit of their ratio. If the limit is 0 or ∞, it indicates one function grows significantly faster than the other. [cite: 1476]

The graph of a function g(x) has a horizontal asymptote at y = -3. Which of the following limit statements must be true?

A) lim x→-3 g(x) = ∞

B) lim x→∞ g(x) = -3 or lim x→-∞ g(x) = -3

C) lim x→0 g(x) = -3

D) lim x→∞ g(x) = 0 and lim x→-∞ g(x) = -3

Correct Answer: B

A horizontal asymptote at y = L is defined by the end behavior of the function. This means that as x approaches either positive infinity or negative infinity (or both), the function's value approaches L. [cite: 1442, 1474]

Given the hierarchy of function growth rates, what is the value of lim x→∞ (ln(x) / x²)?

A) ∞

B) 1

C) 2

D) 0

Correct Answer: D

Polynomial functions, like x², grow much faster than logarithmic functions, like ln(x). When comparing their relative magnitudes using a limit of their ratio, the denominator grows infinitely faster than the numerator, causing the fraction to approach 0. [cite: 1476]

Which statement best describes the behavior of a function f(x) if it is known that lim x→-∞ f(x) = ∞?

A) As x approaches 0, the values of f(x) increase without bound.

B) The graph of f(x) has a horizontal asymptote as x decreases without bound.

C) As x decreases without bound, the values of f(x) also decrease without bound.

D) As x decreases without bound, the values of f(x) increase without bound.

Correct Answer: D

This limit describes the end behavior of the function on the far left of the graph. The notation x→-∞ indicates that x is decreasing without bound, and the result = ∞ indicates that the function's output values are increasing without bound. [cite: 1440, 1469]

Consider the functions f(x) = 100x¹⁰, g(x) = (1.1)ˣ, and h(x) = 1000 ln(x). Based on a comparison of their relative magnitudes, which function will have the largest value as x→∞?

A) f(x)

B) g(x)

C) h(x)

D) All three functions approach ∞ at the same rate.

Correct Answer: B

Exponential functions (like g(x) = (1.1)ˣ) grow faster than any polynomial function (like f(x) = 100x¹⁰), which in turn grows faster than any logarithmic function (like h(x) = 1000 ln(x)). Therefore, as x→∞, g(x) will eventually become the largest. [cite: 1476]

The extension of the limit concept to include limits at infinity allows mathematicians and scientists to formally analyze what?

A) The continuity of a function at a single point.

B) The long-term trends and steady states of systems.

C) The slope of a tangent line to a curve.

D) The exact value of a function at an undefined point.

Correct Answer: B

Limits at infinity are the mathematical tool for describing end behavior. In applied contexts, this corresponds to analyzing the long-term behavior of a system, such as whether a population stabilizes or a chemical reaction reaches equilibrium. [cite: 1472, 1474]

What is the end behavior of the function f(x) = (5eˣ + 4x) / (2eˣ - x³)?

A) The function has a horizontal asymptote at y = 0.

B) The function has a horizontal asymptote at y = 5/2.

C) The function increases without bound as x→∞.

D) The function has a horizontal asymptote at y = -4.

Correct Answer: B

To find the end behavior as x→∞, we compare the relative magnitudes of the terms. The exponential term eˣ grows much faster than any polynomial term (4x or -x³). Therefore, the limit is determined by the ratio of the dominant terms: lim x→∞ (5eˣ / 2eˣ) = 5/2. This means there is a horizontal asymptote at y = 5/2. [cite: 1476]

If a function f(x) has a horizontal asymptote at y=L, what can be concluded about the limit of the function's rate of change, lim x→∞ f'(x), assuming it exists?

A) lim x→∞ f'(x) = L

B) lim x→∞ f'(x) = 1

C) lim x→∞ f'(x) = 0

D) The limit of f'(x) cannot be determined.

Correct Answer: C

If a function approaches a horizontal asymptote y=L, its graph is becoming flat. A flat graph has a slope of zero. Therefore, the function's rate of change, f'(x), must be approaching 0 as x approaches infinity. This is an interpretation of the behavior of a function based on its limit at infinity. [cite: 1440, 1476]