$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

$$\lim_{x \to c} [f(x) + g(x)] = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)$$

Question 1

Given the piecewise function represented in the first image, at which value of $$x is $$the function discontinuous?

Accepted Answer

$$x = 2$$

Question 2

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

Accepted Answer

$$f'(x) = \lim_{h 	o 0} \frac{f(x+h) - f(x)}{h}$$

Question 3

For the function $$f(x) = \frac{1}{(x-4)^2$}$$, what is $$\lim_{x 	o 4} f(x)$$?

Accepted Answer

$$\infty$$

Question 4

Which of the following is NOT a condition for a function to be continuous at $$x = a$$?

Accepted Answer

$$f'(a)$$ exists

Question 5

Suppose $$f(x) is a $$polynomial. Which of the following statements is true?

Accepted Answer

$$f(x) is $$continuous everywhere

Question 6

Given $$f(x) = 2x^2$ + 4x - 5$$, find $$\lim_{x 	o -2} f(x).$$

Accepted Answer

$$-13$$

Question 7

Which scenario is illustrated by the third image (open circle at $$x=a on$$ $$f(x)$$)?

Accepted Answer

A removable discontinuity

Question 8

If $$f(x) = \sqrt{x} - 3 / (x - 9)$$, what is $$\lim_{x 	o 9} f(x)$$?

Accepted Answer

$$\frac{1}{6}$$

Question 9

Which of the following best describes the meaning of the derivative at a point?

Accepted Answer

The slope of the tangent line to the graph at that point

Question 10

For the function $$f(x) = \frac{8x^2$ - 3x - 5}{5x^2$ + 3x + 3}$$, what is $$\lim_{x 	o \infty} f(x)$$?

Accepted Answer

$$\frac{8}{5}$$

Question 11

Which of the following is a reason a function may fail to be differentiable at $$x = a$$?

Accepted Answer

The function has a corner at $$x = a$$

Question 12

Given $$h(t) = 8t - 16t^2$, what is the average rate of change on $$[\frac{1}{2}, 1]$$?

Accepted Answer

$$-12$$

Question 13

Which image best represents a function with a vertical tangent line at $$x = a$$?

Accepted Answer

The fifth image (cusp or vertical tangent)

Question 14

If $$f(x) = 4x - 1 / (x - 3)$$, what is $$\lim_{x 	o 2} f(x)$$?

Accepted Answer

$$-9$$

Question 15

Which of the following is the correct formula for the average rate of change of $$f(x) on$$ $$[x_1$, x_2$]$$?

Accepted Answer

$$\frac{f(x_2$) - f(x_1$)}{x_2$ - x_1$}$$

Question 16

For the function $$f(x) = (6x + 3)^3$, what is $$\lim_{x 	o -1} f(x)$$?

Accepted Answer

$$(-3)^3 = -27$

Question 17

Which of the following is true about rational functions?

Accepted Answer

They are continuous everywhere except where the denominator is zero

Question 18

If $$f(x) = 8$$, what is $$\lim_{x 	o 6} f(x)$$?

Accepted Answer

$$8$$

Question 19

Which of the following is the correct limit property for the sum of two functions?

Accepted Answer

$$\lim_{x 	o c} [f(x) + g(x)] = \lim_{x 	o c} f(x) + \lim_{x 	o c} g(x)$$

Business Calculus Study Notes: Limits, Continuity, and the Derivative

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