Q: Which of the following is the correct definition of a horizontal asymptote for a function $$f(x)$$?

A line $$y = L$$ such that $$\lim\limits_{x \to \infty} f(x) = L or$$ $$\lim\limits_{x \to -\infty} f(x) = L.$$

Question 1

Which of the following statements best describes a vertical asymptote of a function $$f(x)$$?

Accepted Answer

It is a line $$x = a$$ where $$f(x)$$ increases or decreases without bound as $$x$$ approaches $$a.$$

Question 2

Evaluate $$\lim\limits_{x 	o \infty} \frac{2x^2$ + 1}{x^2$}.$$

Accepted Answer

$$2$$

Question 3

Given $$f(x) = \frac{1}{x^2$}$$, what happens to $$f(x) as$$ $$x$$ approaches infinity?

Accepted Answer

$$f(x)$$ approaches $$0$$

Question 4

Which of the following is the correct definition of a horizontal asymptote for a function $$f(x)$$?

Accepted Answer

A line $$y = L$$ such that $$\lim\limits_{x 	o \infty} f(x) = L or$$ $$\lim\limits_{x 	o -\infty} f(x) = L.$$

Question 5

Find the vertical asymptotes of $$f(x) = \frac{x^2$ - 9x + 14}{x^2$ - 5x + 6}.$$

Accepted Answer

$$x = 2$$ and $$x = 3$$

Question 6

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

Accepted Answer

$$\infty$$

Question 7

For $$f(x) = 2x^4$ + 3x^2$ - 12$$, what is $$\lim\limits_{x 	o -\infty} f(x)$$?

Accepted Answer

$$-\infty$$

Question 8

Which of the following describes the end behavior of $$f(x) = x^n as$$ $$x 	o \infty$$ when $$n is $$even?

Accepted Answer

$$f(x) 	o \infty$$

Question 9

Evaluate $$\lim\limits_{x 	o \infty} \frac{5x^7$ + x - 1}{3x^7$ - 2}.$$

Accepted Answer

$$\frac{5}{3}$$

Question 10

Suppose $$f(x) = \frac{1 - 3x^4$ + 2x^9$}{4x^9$ - x^5$ + 5x^6$}. $$What is $$\lim\limits_{x 	o \infty} f(x)$$?

Accepted Answer

$$\frac{2}{4} = \frac{1}{2}$$

Question 11

Which of the following is true about the function $$f(x) = \frac{x}{(x^2$ - 1)^2$} as$$ $$x 	o 1^-$$?

Accepted Answer

$$f(x) 	o -\infty$$

Question 12

Given $$f(x) = 2 + \frac{1}{x}$$, what is $$\lim\limits_{x 	o -\infty} f(x)$$?

Accepted Answer

$$2$$

Question 13

If $$\lim\limits_{x 	o a^-} f(x) = \infty$$ and $$\lim\limits_{x 	o a^+} f(x) = -\infty$$, what does this indicate about the graph of $$f(x) at$$ $$x = a$$?

Accepted Answer

There is a vertical asymptote at $$x = a.$$

Question 14

Evaluate $$\lim\limits_{x 	o \infty} \frac{x^8$ + 5}{12 - x^3$ - x$$^{10}$$}.$$

Accepted Answer

$$0$$

Question 15

Which of the following is the correct end behavior for $$f(x) = -2x^3$ as x$$ 	o -\infty$$?

Accepted Answer

$$f(x) 	o \infty$$

Question 16

Find $$\lim\limits_{x 	o \infty} \frac{3x^4$ - 6x^2$ + x - 10}{x^4$}.$$

Accepted Answer

$$3$$

Question 17

A rational function $$f(x) = \frac{p(x)}{q(x)}$$ has a horizontal asymptote at $$y = 0 if$$:

Accepted Answer

The degree of $$p(x) is $$less than the degree of $$q(x).$$

Question 18

Evaluate $$\lim\limits_{x 	o 1^-} \frac{x}{(x^2$ - 1)^2$}.$$

Accepted Answer

$$-\infty$$

Question 19

If $$f(x) = \frac{1}{x}$$, what is $$\lim\limits_{x 	o 0^+} f(x)$$?

Accepted Answer

$$\infty$$

Limits at Infinity and Asymptotes: A Calculus Study Guide

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