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AP Calculus BC 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

Which aspect of a function's behavior is described by a limit at infinity?

All Questions (10)

Which aspect of a function's behavior is described by a limit at infinity?

A) The function's value at x=0

B) The location of a vertical asymptote

C) The end behavior of the function

D) The slope of the function at a specific point

Correct Answer: C

According to the provided content, limits at infinity are used to describe the end behavior of a function, which is how the function behaves as its input `x` approaches positive or negative infinity.

If it is known that `lim x→∞ f(x) = 5`, what does this imply about the graph of `y = f(x)`?

A) The graph has a vertical asymptote at x = 5.

B) The graph has a horizontal asymptote at y = 5.

C) The graph has a y-intercept at (0, 5).

D) The function's value is 5 for all x > 0.

Correct Answer: B

A finite limit at infinity corresponds to a horizontal asymptote. The statement `lim x→∞ f(x) = 5` means that the function's values get arbitrarily close to 5 as x increases without bound, which defines the horizontal asymptote y = 5.

How does the concept of a 'limit at infinity' extend the traditional concept of a limit at a finite point `c`?

A) It allows for the analysis of function behavior at points where the function is undefined.

B) It is used exclusively for functions that are infinite everywhere.

C) It shifts the focus from a function's behavior near a specific point to its overall long-term trend or end behavior.

D) It is a method to find the exact maximum or minimum value of a function.

Correct Answer: C

The concept of a limit is extended to infinity to analyze the end behavior of functions. While a limit at a finite point `c` describes behavior as `x` approaches `c`, a limit at infinity describes the behavior as `x` grows without bound.

Given two functions `f(x)` and `g(x)` where `lim x→∞ [f(x) / g(x)] = 0`. What can be concluded about the relative magnitudes of `f(x)` and `g(x)` as `x` approaches infinity?

A) The magnitude of `f(x)` grows faster than the magnitude of `g(x)`.

B) The magnitude of `g(x)` grows faster than the magnitude of `f(x)`.

C) `f(x)` and `g(x)` grow at the same rate.

D) Both `f(x)` and `g(x)` must approach 0.

Correct Answer: B

Limits can be used to compare the relative magnitudes of functions. If the limit of the ratio `f(x)/g(x)` is 0, it means the denominator, `g(x)`, is increasing in magnitude much more rapidly than the numerator, `f(x)`.

The end behavior of a function `g(x)` is described by the limit statement `lim x→-∞ g(x) = -2`. Which of the following provides the correct graphical interpretation of this statement?

A) The graph of `g(x)` has an x-intercept at (-2, 0).

B) The graph of `g(x)` has a vertical asymptote at x = -2.

C) The graph of `g(x)` approaches the horizontal line y = -2 as x decreases without bound.

D) The graph of `g(x)` approaches the point (0, -2).

Correct Answer: C

Limits at infinity describe end behavior. A limit as x approaches negative infinity describes the behavior on the far left of the graph. A finite limit of -2 means the function values approach the horizontal line y = -2.

If `lim x→∞ [h(x) / k(x)] = ∞`, what does this imply about the relationship between the rates of change and magnitudes of `h(x)` and `k(x)` for very large values of `x`?

A) The magnitude of `h(x)` is growing significantly faster than the magnitude of `k(x)`.

B) The magnitude of `k(x)` is growing significantly faster than the magnitude of `h(x)`.

C) Both functions are growing at a comparable rate.

D) The function `k(x)` must be approaching a constant value.

Correct Answer: A

Using limits to compare the relative magnitudes of functions, if the limit of their ratio is infinity, it indicates that the numerator, `h(x)`, is growing without bound much more rapidly than the denominator, `k(x)`.

A function `f(x)` is found to have two different horizontal asymptotes, `y = 10` and `y = -10`. Which pair of limit statements must be true to describe this end behavior?

A) `lim x→10 f(x) = ∞` and `lim x→-10 f(x) = -∞`

B) `lim x→∞ f(x) = 10` and `lim x→-∞ f(x) = -10` (or vice-versa)

C) `lim x→0 f(x) = 10` and `lim x→0 f(x) = -10`

D) `lim x→∞ f(x) = ∞` and `lim x→-∞ f(x) = -∞`

Correct Answer: B

Horizontal asymptotes are determined by the limits of the function as x approaches positive and negative infinity. To have two distinct horizontal asymptotes, the limit at positive infinity must be a different finite value from the limit at negative infinity.

Let `f(x)` and `g(x)` be two functions such that `lim x→∞ f(x) = L` and `lim x→∞ g(x) = M`, where `L` and `M` are finite, non-zero constants. The evaluation of `lim x→∞ [f(x) / g(x)]` allows for a comparison of what property of the two functions?

A) Their y-intercepts.

B) The location of their vertical asymptotes.

C) The relative magnitudes of their end behavior.

D) Their local maximum and minimum values.

Correct Answer: C

The limit of the ratio will be L/M. Because this result is a finite, non-zero number, it indicates that as x approaches infinity, the functions f(x) and g(x) have comparable magnitudes. This is a direct application of using limits to compare the relative magnitudes and rates of change of functions.

The population of a bacterial culture is modeled by the function `P(t)`, where `t` is time in hours. If a scientist determines that `lim t→∞ P(t) = 1,000,000`, what is the best interpretation of this result?

A) The population will reach exactly 1,000,000 and then immediately die off.

B) The initial population of the culture was 1,000,000.

C) The population's growth rate is 1,000,000 bacteria per hour.

D) The population approaches a stable, long-term carrying capacity of 1,000,000.

Correct Answer: D

This is an application of interpreting the behavior of a function using a limit at infinity. The limit describes the end behavior or long-term trend. In this biological context, a finite limit represents the carrying capacity of the environment, which is the stable population size the culture approaches over time.

The mathematical tool used to formally define and analyze the end behavior of a function is:

A) The derivative at a point.

B) The definite integral.

C) The concept of continuity.

D) The limit at infinity.

Correct Answer: D

The provided content explicitly states that limits at infinity describe end behavior. This concept was extended from limits at a finite point specifically to provide a rigorous way to analyze how functions behave as their inputs become arbitrarily large or small.