Q: If $$\lim_{x \to 1^-} H(x) = 4$$ and $$\lim_{x \to 1^+} H(x) = 2$$, what can you conclude about $$\lim_{x \to 1} H(x)$$?

$$\lim_{x \to 1} H(x)$$ does not exist.

Question 1

Which of the following statements best describes the meaning of the limit $$\lim_{x 	o a} f(x) = L$$?

Accepted Answer

As $$x$$ approaches $$a$$, $$f(x)$$ gets arbitrarily close to $$L$$, but $$x$$ does not have to equal $$a.$$

Question 2

Given the function $$f(x) = \frac{x^2$ - 9}{x - 3}$$, what is $$\lim_{x 	o 3} f(x)$$?

Accepted Answer

$$6$$

Question 3

Suppose $$H(x) is $$defined as $$H(x) = x^2$ + 2$$ for $$x \u003c 1$$ and $$H(x) = 4 - x$$ for $$x \geq 1. $$What is $$\lim_{x 	o 1^-} H(x)$$?

Accepted Answer

$$4$$

Question 4

For the same function $$H(x) as $$above, what is $$\lim_{x 	o 1^+} H(x)$$?

Accepted Answer

$$2$$

Question 5

If $$\lim_{x 	o 1^-} H(x) = 4$$ and $$\lim_{x 	o 1^+} H(x) = 2$$, what can you conclude about $$\lim_{x 	o 1} H(x)$$?

Accepted Answer

$$\lim_{x 	o 1} H(x)$$ does not exist.

Question 6

Which of the following is NOT a requirement for a function $$f(x) to be $$continuous at $$x = a$$?

Accepted Answer

$$f(x) is $$differentiable at $$x = a.$$

Question 7

Evaluate $$\lim_{x 	o \infty} \left( \frac{1}{x-2} + 3 ight).$$

Accepted Answer

$$3$$

Question 8

A company’s profit $$P(x) in $$thousands of dollars is given by $$P(x) = 2x^2$ + 3x. $$What is the average rate of change of profit as production increases from $$x = 1 to$$ $$x = 4$$?

Accepted Answer

$$15$$

Question 9

Which of the following is the correct simplified form of the difference quotient for $$f(x) = x^2$?

Accepted Answer

$$2x + h$$

Question 10

If $$f(x) = x^3$, what is the derivative f$$'(x)$$ using the power rule?

Accepted Answer

$$3x^2$$

Question 11

For $$f(x) = x^3$ - 6x^2$, at which x$-values is the tangent line horizontal?

Accepted Answer

$$x = 0$$ and $$x = 4$$

Question 12

Which of the following is NOT a reason why a function might fail to be differentiable at a point?

Accepted Answer

The function is linear at the point.

Question 13

If $$y = 7x^4$ - 3x + 2$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$28x^3 - 3$

Question 14

Suppose $$f(x) = |x - 6|. At $$which value is $$f(x)$$ NOT differentiable?

Accepted Answer

$$x = 6$$

Question 15

If $$f(x) = 3x^2$ + 2x$$, what is the slope of the tangent line at $$x = -2$$?

Accepted Answer

$$-8$$

Question 16

Which of the following is the correct application of the sum-difference rule for differentiation?

Accepted Answer

$$\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$$

Question 17

A function $$f(x) is $$defined as $$f(x) = \frac{x^2$ - 25}{x - 5}$$ for $$x 
eq 5$$ and $$f(5) = c. $$What value of $$c$$ makes $$f(x)$$ continuous at $$x = 5$$?

Accepted Answer

$$10$$

Question 18

If $$y = 3x^5$ - 2x^3$ + 7$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$15x^4$$ - $$6x^2$$

Question 19

Which of the following best describes the average rate of change of a function $$f(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

$$\frac{f(b) - f(a)}{b - a}$$

Business Calculus: Limits, Continuity, and Differentiation

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