Q: For the function $$f(x) = x^2$ - 2$$ for $$x \geq 1$$ and $$f(x) = -3 + 2x$$ for $$x \u003c 1$$, is $$f(x)$$ continuous at $$x = 1$$?

Yes, because $$\lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1)$$

Q: Which of the following statements about limits is correct?

If $$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$$, then $$\lim_{x \to a} f(x)$$ exists

Q: Which of the following is the correct definition of the derivative of $$f(x) at$$ $$x = a$$?

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

Question 1

Suppose $$f(x) = x^2$ - 2$$ for $$x \geq 1$$ and $$f(x) = -3 + 2x$$ for $$x \u003c 1. $$What is $$\lim_{x 	o 1^-} f(x)$$?

Accepted Answer

$$-1$$

Question 2

For the function $$f(x) = x^2$ - 2$$ for $$x \geq 1$$ and $$f(x) = -3 + 2x$$ for $$x \u003c 1$$, is $$f(x)$$ continuous at $$x = 1$$?

Accepted Answer

Yes, because $$\lim_{x 	o 1^-} f(x) = \lim_{x 	o 1^+} f(x) = f(1)$$

Question 3

Given the function $$f(x) = \frac{1}{x}$$, construct a table of $$f(x)$$ values for $$x = 0.9, 0.99, 0.999, 1.001, 1.01, 1.1. $$What does the table suggest about $$\lim_{x 	o 1} f(x)$$?

Accepted Answer

The limit is $$1$$ because $$f(x)$$ approaches $$1$$ from both sides

Question 4

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

Accepted Answer

$$2$$

Question 5

A business's revenue function is $$R(x) = 5x^2$ + 3x$$, where $$x is $$the number of units sold. What is the marginal revenue when $$x = 10$$?

Accepted Answer

$$103$$

Question 6

Given the function $$f(x) = \sqrt{x}$$, what is the derivative $$f'(x) at$$ $$x = 4$$?

Accepted Answer

$$\frac{1}{4}$$

Question 7

Which of the following statements about limits is correct?

Accepted Answer

If $$\lim_{x 	o a^-} f(x) = \lim_{x 	o a^+} f(x)$$, then $$\lim_{x 	o a} f(x)$$ exists

Question 8

A cost function is $$C(x) = 100 + 20x. $$What is the marginal cost at $$x = 5$$?

Accepted Answer

$$20$$

Question 9

For the function $$f(x) = \frac{1}{x-2}$$, at which value of $$x is $$the function discontinuous?

Accepted Answer

$$x = 2$$

Question 10

Suppose $$f(x) is $$continuous everywhere except at $$x = 3. $$Which of the following could be a reason for the discontinuity?

Accepted Answer

$$f(3) is $$not defined

Question 11

If $$f(x) = \frac{x^2$ - 4}{x - 2}$$, what is $$\lim_{x 	o 2} f(x)$$?

Accepted Answer

$$2$$

Question 12

A company’s profit function is $$P(x) = 10x - x^2$. What is the marginal profit at x = 3$?

Accepted Answer

$$7$$

Question 13

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

Accepted Answer

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

Question 14

If $$f(x) = 3x^2$ + 2x$$, what is $$f'(x)$$?

Accepted Answer

$$6x + 2$$

Question 15

A function $$f(x) is $$defined as $$f(x) = \frac{1}{x}$$ for $$x 
eq 0. Is$$ $$f(x)$$ continuous at $$x = 0$$?

Accepted Answer

No, because $$f(0) is $$not defined

Question 16

If $$f(x) = x^2$ and g(x) = 2x$, what is the derivative of f(x) + g(x$$)$$?

Accepted Answer

$$2x + 2$$

Question 17

Suppose $$f(x) is $$defined as $$f(x) = \frac{x^2$ - 9}{x - 3}$$ for $$x 
eq 3. $$What is $$\lim_{x 	o 3} f(x)$$?

Accepted Answer

$$6$$

Question 18

A function $$f(x) is $$defined as $$f(x) = \frac{1}{x-4}. At $$which value of $$x is$$ $$f(x)$$ discontinuous?

Accepted Answer

$$x = 4$$

Question 19

If $$f(x) = x^2$ - 2$$ for $$x \geq 1$$ and $$f(x) = -3 + 2x$$ for $$x \u003c 1$$, what is $$f(1)$$?

Accepted Answer

$$-1$$

Business Calculus Limits, Derivatives, and Continuity Study Guide

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