The function approaches $$y = 5 as$$ $$x \to \infty.$$

Question 1

Which of the following statements best describes the horizontal asymptote of the function $$f(x) = 5 - \frac{2}{x^2$} as$$ $$x 	o \infty$$?

Accepted Answer

The function approaches $$y = 5 as$$ $$x 	o \infty.$$

Question 2

Evaluate the limit: $$\lim_{x 	o \infty} \left(5 + \frac{\sin x}{\sqrt{x}}ight).$$

Accepted Answer

$$5$$

Question 3

Which theorem is used to show that $$\lim_{x 	o \infty} \frac{\sin x}{\sqrt{x}} = 0$$?

Accepted Answer

Squeeze Theorem

Question 4

What is the end behavior of the function $$p(x) = 3x^4$ - 6x^2$ + x - 10 as$$ $$x 	o \infty$$?

Accepted Answer

$$p(x) 	o \infty$$

Question 5

For the function $$q(x) = -2x^3$ + 3x^2$ - 12$$, what is $$\lim_{x 	o -\infty} q(x)$$?

Accepted Answer

$$-\infty$$

Question 6

Evaluate $$\lim_{x 	o \infty} \frac{3x + 2}{x^2$ - 1}.$$

Accepted Answer

$$0$$

Question 7

What is the horizontal asymptote of $$g(x) = \frac{40x^4$ + 4x^2$ - 1}{10x^4$ + 8x^2$ + 1}$$?

Accepted Answer

$$y = 4$$

Question 8

Determine the limit: $$\lim_{x 	o \infty} \frac{x^3$ - 2x + 1}{2x + 4}.$$

Accepted Answer

$$\infty$$

Question 9

For the function $$f(x) = \frac{2x^2$ + 6x - 2}{x + 1}$$, what type of asymptote does the graph have?

Accepted Answer

Slant (oblique) asymptote

Question 10

Evaluate $$\lim_{x 	o \infty} \frac{10x^3$ - 3x^2$ + 8}{\sqrt{25x^6$ + x^4$ + 2}}.$$

Accepted Answer

$$2$$

Question 11

What is the limit $$\lim_{x 	o -\infty} \frac{10x^3$ - 3x^2$ + 8}{\sqrt{25x^6$ + x^4$ + 2}}$$?

Accepted Answer

$$-2$$

Question 12

Which of the following functions does NOT have a limit as $$x 	o \infty$$?

Accepted Answer

$$f(x) = \cos x$$

Question 13

What is the end behavior of $$e^x as$$ $$x 	o -\infty$$?

Accepted Answer

$$e^x 	o 0$$

Question 14

For $$f(x) = 4 - \ln 3x$$, what is $$\lim_{x 	o \infty} f(x)$$?

Accepted Answer

$$-\infty$$

Question 15

Which statement is true about the horizontal asymptotes of rational functions?

Accepted Answer

A rational function can cross its horizontal asymptote.

Question 16

If the degree of the numerator is greater than the degree of the denominator by exactly one, what type of asymptote does the rational function have?

Accepted Answer

Slant (oblique) asymptote

Question 17

What is the limit $$\lim_{x 	o \infty} \ln x$$?

Accepted Answer

$$\infty$$

Question 18

Which of the following best describes the behavior of $$f(x) = \frac{1}{x^n} as$$ $$x 	o \infty$$, where $$n is a $$positive integer?

Accepted Answer

$$f(x) 	o 0$$

Question 19

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

Accepted Answer

$$y = 3$$

2.5 Limits at Infinity and End Behavior

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