$$f(x) is $$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

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