Question 1

Which of the following best describes the difference quotient for a function $f(x)$?

Accepted Answer

It is the expression $\frac{f(x+h) - f(x)}{h}$ where $h$ is a variable.

Question 2

What is the derivative $f'(x)$ of $f(x) = x^2$ using the limit definition?

Accepted Answer

$2x$

Question 3

For $f(x) = \frac{1}{x}$, what is $f'(x)$ using the difference quotient and limit?

Accepted Answer

$-\frac{1}{x^2}$

Question 4

Consider the function $f(x) = \frac{1}{x-1}$. What happens to $f(x)$ as $x$ approaches $1$ from the left?

Accepted Answer

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

Question 5

Which statement 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 6

For $f(x) = \frac{1}{x^2}$, what is $\lim_{x 	o 0^-} f(x)$?

Accepted Answer

$\infty$

Question 7

Given $h(x) = \frac{x^2 + 1}{x^2 - 4}$, at which values of $x$ does $h(x)$ have vertical asymptotes?

Accepted Answer

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

Question 8

For $h(x) = \frac{x^2 + 1}{x^2 - 4}$, what is $\lim_{x 	o 2^-} h(x)$?

Accepted Answer

$-\infty$

Question 9

For $h(x) = \frac{x^2 + 1}{x^2 - 4}$, what is $\lim_{x 	o 2^+} h(x)$?

Accepted Answer

$\infty$

Question 10

Which of the following is a correct condition for a vertical asymptote of $y = \frac{f(x)}{g(x)}$ at $x = a$?

Accepted Answer

$g(a) = 0$ and $f(a) 
eq 0$

Question 11

What is the behavior of $\lim_{x 	o 0^+} \ln(x)$?

Accepted Answer

$-\infty$

Question 12

A pharmaceutical company models the concentration of a drug in the bloodstream as $C(t) = \frac{1}{t-3}$, where $t$ is time in hours. What happens to the concentration as $t$ approaches $3$ from above?

Accepted Answer

The concentration approaches $\infty$

Question 13

Suppose $f(x) = \sqrt{x}$ and $g(x) = x(x+2)$. Evaluate $\lim_{x 	o 0^+} \frac{\sqrt{x}}{x(x+2)}$ after simplifying.

Accepted Answer

$\infty$

Question 14

Which of the following statements about $y = \ln(x)$ is correct?

Accepted Answer

The line $x = 0$ is a vertical asymptote.

Question 15

A business models its profit as $P(x) = \frac{x^2 + 1}{x^2 - 4}$, where $x$ is the number of units sold. What happens to the profit as $x$ approaches $-2$ from the right?

Accepted Answer

Profit approaches $-\infty$

Question 16

If $f(x) = \frac{1}{x-1}$, which of the following best describes the vertical asymptote?

Accepted Answer

The vertical line $x = 1$ is a vertical asymptote.

Question 17

For the function $f(x) = \frac{1}{x^2}$, what is the behavior as $x$ approaches $0$ from either side?

Accepted Answer

The function approaches $\infty$ from both sides.

Question 18

A company models its cost function as $C(x) = \frac{1}{x-5}$, where $x$ is the number of products produced. What happens to the cost as $x$ approaches $5$ from below?

Accepted Answer

Cost approaches $-\infty$

Question 19

Which of the following is a correct method to determine the sign of a limit near a vertical asymptote for a rational function?

Accepted Answer

Choose test values near the asymptote and analyze the sign of the numerator and denominator.

Difference Quotients, Infinite Limits, and Vertical Asymptotes in Business Calculus

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