Question 1

Which of the following best describes a horizontal asymptote for a function $$f(x)$$?

Accepted Answer

A line $$y = L$$ that the graph of $$f(x)$$ approaches as $$x 	o \infty or$$ $$x 	o -\infty.$$

Question 2

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

Accepted Answer

$$2$$

Question 3

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

Accepted Answer

$$0$$

Question 4

Which of the following statements is TRUE about the end behavior of $$f(x) = \frac{3x^3$ + 2}{x^2$ - 1}$$?

Accepted Answer

There is no horizontal asymptote.

Question 5

For the function $$f(x) = \frac{4x^2$ - 1}{2x^2$ + 3}$$, what is the horizontal asymptote?

Accepted Answer

$$y = 2$$

Question 6

If $$\lim\limits_{x 	o \infty} f(x) = L$$, what does this mean about the behavior of $$f(x) as$$ $$x$$ increases without bound?

Accepted Answer

$$f(x)$$ approaches the value $$L as$$ $$x$$ becomes very large.

Question 7

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

Accepted Answer

$$\frac{5}{2}$$

Question 8

Which of the following functions has a horizontal asymptote at $$y = 0$$?

Accepted Answer

$$f(x) = \frac{2x}{x^2$ + 1}$$

Question 9

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 10

Which of the following best describes the end behavior of $$f(x) = x^4$ as x$$ 	o -\infty$$?

Accepted Answer

$$f(x) 	o \infty$$

Question 11

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

Accepted Answer

$$\frac{7}{2}$$

Question 12

Which of the following is a correct application of the Squeeze Theorem for $$\lim\limits_{x 	o \infty} \frac{\sin x}{x}$$?

Accepted Answer

Since $$-1 \leq \sin x \leq 1$$, $$-\frac{1}{x} \leq \frac{\sin x}{x} \leq \frac{1}{x}$$, and both bounds go to $$0 as$$ $$x 	o \infty.$$

Question 13

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

Accepted Answer

$$0$$

Question 14

Which of the following describes the horizontal asymptote of $$f(x) = \frac{5x^2$ - 1}{x^2$ + 3}$$?

Accepted Answer

$$y = 5$$

Question 15

If $$f(x) = \frac{x^2$ + 1}{x^3$ + 2}$$, what is the end behavior as $$x 	o \infty$$?

Accepted Answer

$$f(x) 	o 0$$

Question 16

Which of the following is true about the function $$f(x) = \frac{x^2$ - 4}{x^2$ + 4}$$?

Accepted Answer

It has a horizontal asymptote at $$y = 1.$$

Limits at Infinity and Horizontal Asymptotes

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