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

$$g(a) = 0$$ and $$f(a) \neq 0$$

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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