$f(x)$ is continuous at $x = a$ if $\lim_{x \to a} f(x)$ exists and equals $f(a)$.

$\lim_{x \to a} f(x)$ does not exist

$\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

Question 1

Which of the following best describes a one-sided limit from the right at $x = a$?

Accepted Answer

It is the value $f(x)$ approaches as $x$ approaches $a$ from values greater than $a$.

Question 2

Given $\lim_{x 	o 2} (3x^2 - 2x + 1)$, what is the value of the limit?

Accepted Answer

$11$

Question 3

If $\lim_{x 	o a} f(x) = L$ and $\lim_{x 	o a} g(x) = M$, what is $\lim_{x 	o a} [f(x) + g(x)]$?

Accepted Answer

$L + M$

Question 4

Suppose $f(x)$ and $g(x)$ are polynomials. What is $\lim_{x 	o a} [f(x) \cdot g(x)]$?

Accepted Answer

$f(a) \cdot g(a)$

Question 5

Evaluate $\lim_{x 	o 0} \frac{|x|}{x}$.

Accepted Answer

The limit does not exist (DNE)

Question 6

Which of the following is true about the continuity of $f(x)$ at $x = a$?

Accepted Answer

$f(x)$ is continuous at $x = a$ if $\lim_{x 	o a} f(x)$ exists and equals $f(a)$.

Question 7

Find the derivative of $f(x) = 5x^3 - 2x + 4$.

Accepted Answer

$15x^2 - 2$

Question 8

If $C(x)$ is the cost function, what does the marginal cost $C'(x)$ represent?

Accepted Answer

The cost of producing one more unit at $x$ units produced

Question 9

A factory's cost function is $C(x) = 200 + 20x$. What is the marginal cost?

Accepted Answer

$20$

Question 10

If the price-demand function is $p = 60 - 5x$, what is the revenue function $R(x)$?

Accepted Answer

$R(x) = 60x - 5x^2$

Question 11

Given $C(x) = 200 + 20x$ and $R(x) = 60x - 5x^2$, what is the profit function $P(x)$?

Accepted Answer

$P(x) = 60x - 5x^2 - 200 - 20x$

Question 12

If $C(x) = 200 + 20x$ and $R(x) = 60x - 5x^2$, at what value of $x$ does break-even occur?

Accepted Answer

When $R(x) = C(x)$

Question 13

Suppose $f(x)$ is not continuous at $x = a$. Which of the following could be a reason?

Accepted Answer

$\lim_{x 	o a} f(x)$ does not exist

Question 14

Find the equation of the tangent line to $f(x) = x^2$ at $x = 3$.

Accepted Answer

$y = 6x - 9$

Question 15

If $f(x)$ is differentiable at $x = a$, which of the following must be true?

Accepted Answer

$f(x)$ is continuous at $x = a$

Question 16

A company finds that increasing production from 150 to 151 units increases revenue by approximately $30. What concept does this illustrate?

Accepted Answer

Marginal revenue

Question 17

If $f(x)$ has a horizontal asymptote at $y = L$, what does this mean?

Accepted Answer

$f(x)$ approaches $L$ as $x$ goes to infinity or negative infinity

Question 18

Which of the following is the correct definition of the instantaneous rate of change of $f(x)$ at $x = a$?

Accepted Answer

$\lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$

Limits and Limit Rules in Business Calculus

Study Guide - Practice Questions