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

$$q(a) = 0$$ and $$p(a) \neq 0$$

Question 1

Given the function $$f(x) = \frac{1}{x-1}$$, what is the behavior of $$f(x) as$$ $$x$$ approaches 1 from the right ($$x 	o 1^+$$)?

Accepted Answer

$$f(x)$$ approaches $$\infty$$

Question 2

Given the function $$f(x) = \frac{1}{x-1}$$, what is the behavior of $$f(x) as$$ $$x$$ approaches 1 from the left ($$x 	o 1^-$$)?

Accepted Answer

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

Question 3

Which of the following statements is true about the limit $$\lim_{x 	o 1} \frac{1}{x-1}$$?

Accepted Answer

The limit does not exist because the left and right limits are not equal

Question 4

What is the vertical asymptote of the function $$f(x) = \frac{1}{x-1}$$?

Accepted Answer

$$x = 1$$

Question 5

Which of the following best describes the use of the infinity symbol $$\infty in $$limits?

Accepted Answer

It describes the manner in which a limit fails to exist due to unbounded growth.

Question 6

For the rational function $$f(x) = \frac{2x+3}{x^2$-4}$$, which values of $$x$$ are vertical asymptotes?

Accepted Answer

$$x = 2$$ and $$x = -2$$

Question 7

Consider the polynomial $$p(x) = 5x^3$ - 2x^2$ - 3x + 6. $$What is $$\lim_{x 	o \infty} p(x)$$?

Accepted Answer

$$\infty$$

Question 8

Which of the following statements about polynomial functions is correct?

Accepted Answer

Polynomial functions of degree 0 have horizontal asymptotes.

Question 9

Given $$f(x) = \frac{3x^2$ + 2}{x^2$ - 1}$$, what is $$\lim_{x 	o \infty} f(x)$$?

Accepted Answer

3

Question 10

A company’s profit function is $$P(x) = 7x^3$ - 2x^2$ + 3x - 4. $$Which of the following describes the end behavior as $$x 	o -\infty$$?

Accepted Answer

$$P(x)$$ approaches $$-\infty$$

Question 11

Which of the following is true about the horizontal asymptote of $$f(x) = \frac{2x}{x+1}$$?

Accepted Answer

$$y = 2$$

Question 12

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

Accepted Answer

0

Question 13

Which of the following best describes a vertical asymptote?

Accepted Answer

A line $$x = a$$ where the function grows without bound as $$x$$ approaches $$a.$$

Question 14

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

Accepted Answer

The function is undefined at $$x = 1$$

Question 15

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

Accepted Answer

$$q(a) = 0$$ and $$p(a) 
eq 0$$

Question 16

Which of the following best describes the end behavior of $$f(x) = -2x^4$ + 5x^2$ - 3 as$$ $$x 	o \infty$$?

Accepted Answer

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

Question 17

A newly released smart-phone operating system gives users an update notice when they download a new app. If the probability of receiving a notice is modeled by $$P(x) = \frac{1}{x}$$, what happens as the number of downloads $$x$$ increases without bound?

Accepted Answer

$$P(x)$$ approaches 0$

Question 18

Which of the following is NOT a possible behavior for a rational function as $$x 	o \infty$$?

Accepted Answer

Oscillating between two values

Question 19

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

Accepted Answer

$$y = 2$$

Question 20

Which of the following is true about the limit $$\lim_{x 	o -\infty} (7x^3$ - 2x^2$ + 3x - 4)$$?

Accepted Answer

The limit is $$-\infty$$

Infinite Limits and Limits at Infinity: Study Notes for Business Calculus 2.2

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