Q: For the function $f(x) = \frac{1}{x}$, which of the following statements is true?

$\lim\limits_{x \to \infty} f(x) = 0$

Q: Which of the following is true about the function $f(x) = e^{-x}$?

$\lim\limits_{x \to \infty} f(x) = 0$

Q: For the function $f(x) = \frac{1}{x^2}$, which of the following is correct?

$\lim\limits_{x \to \infty} f(x) = 0$

Q: If $f(x) = \frac{anx^n + \ldots + a_0}{bmx^m + \ldots + b_0}$, which of the following is true if $n < m$?

$\lim\limits_{x \to \infty} f(x) = 0$

Q: Which of the following limits is undefined?

$\lim\limits_{x \to \infty} \sin(x)$

Question 1

Which of the following best describes the horizontal asymptote(s) of the function $h(x) = \frac{x^2 + 1}{x^2 - 4}$?

Accepted Answer

There is one horizontal asymptote at $y = 1$.

Question 2

Evaluate $\lim\limits_{x 	o \infty} \frac{3x^2 + 5x - 2}{4x^2 - 3x + 2}$.

Accepted Answer

$\frac{3}{4}$

Question 3

For the function $f(x) = \frac{1}{x}$, which of the following statements is true?

Accepted Answer

$\lim\limits_{x 	o \infty} f(x) = 0$

Question 4

Consider the function $f(x) = \sin(x)$. What is $\lim\limits_{x 	o \infty} f(x)$?

Accepted Answer

The limit does not exist.

Question 5

Given the rational function $f(x) = \frac{2x^3 + 1}{x^2 - 5}$, what is $\lim\limits_{x 	o \infty} f(x)$?

Accepted Answer

$\infty$

Question 6

Which of the following is true about the function $f(x) = e^{-x}$?

Accepted Answer

$\lim\limits_{x 	o \infty} f(x) = 0$

Question 7

Suppose $F = G \frac{m_1 m_2}{r^2}$ represents the gravitational force between two masses. What happens to $F$ as $r 	o \infty$?

Accepted Answer

$F$ approaches $0$

Question 8

For the function $f(x) = \frac{1}{x^2}$, which of the following is correct?

Accepted Answer

$\lim\limits_{x 	o \infty} f(x) = 0$

Question 9

Evaluate $\lim\limits_{x 	o -\infty} \frac{1}{1 + e^{-x}}$.

Accepted Answer

$0$

Question 10

Which of the following best describes the horizontal asymptotes of $f(x) = \frac{1}{1 + e^{-x}}$?

Accepted Answer

There are horizontal asymptotes at $y = 0$ and $y = 1$.

Question 11

Given $P(t) = Ce^{kt}$ with $k < 0$ and $C > 0$, what is $\lim\limits_{t 	o \infty} P(t)$?

Accepted Answer

$0$

Question 12

For the rational function $f(x) = \frac{x^2 - 3x + 2}{5x^3 - 3x^2 + 2x + 2}$, what is $\lim\limits_{x 	o \infty} f(x)$?

Accepted Answer

$0$

Question 13

Which of the following statements about horizontal asymptotes is correct?

Accepted Answer

A function can have two different horizontal asymptotes.

Question 14

If $f(x) = \frac{anx^n + \ldots + a_0}{bmx^m + \ldots + b_0}$, which of the following is true if $n < m$?

Accepted Answer

$\lim\limits_{x 	o \infty} f(x) = 0$

Question 15

Suppose $f(x) = x^3$. What is $\lim\limits_{x 	o -\infty} f(x)$?

Accepted Answer

$-\infty$

Question 16

Which of the following best describes the behavior of $f(x) = e^x$ as $x 	o -\infty$?

Accepted Answer

$f(x)$ approaches $0$

Question 17

For the function $f(x) = \frac{x^2 + 1}{x^2 - 4}$, for which intervals is $f(x) > 1$?

Accepted Answer

For $x < -2$ and $x > 2$

Question 18

Which of the following limits is undefined?

Accepted Answer

$\lim\limits_{x 	o \infty} \sin(x)$

Question 19

If $P(t) = Ce^{kt}$ with $k > 0$, what does $\lim\limits_{t 	o \infty} P(t)$ predict about the modeled quantity?

Accepted Answer

It grows without bound.

Limits at Infinity and Horizontal Asymptotes in Business Calculus

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