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 $$n \u003c m$$?

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

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}$$ with k  0$, what is $$\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 $$n \u003c m$$?

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 $$f(x) \u003e 1$$?

Accepted Answer

For $$x  2$$

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}$$ with k$$ \u003e 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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