$$\displaystyle \lim_{x \to 2} f(x) = 4$$ and $$f(2) = 4$$

$$\displaystyle \lim_{x \to 2} g(x) = 4$$ and $$g(2) is $$not defined

$$\displaystyle \lim_{x \to c} f(x) = f(c)$$

Question 1

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

Accepted Answer

$$f(x) is $$differentiable at $$x = c$$

Question 2

Given $$f(x) = x + 2$$, as shown in the first image, which of the following statements is true?

Accepted Answer

$$\displaystyle \lim_{x 	o 2} f(x) = 4$$ and $$f(2) = 4$$

Question 3

Consider $$g(x) = \dfrac{x^2$ - 4}{x - 2} as $$shown in the second image. Which of the following is correct?

Accepted Answer

$$\displaystyle \lim_{x 	o 2} g(x) = 4$$ and $$g(2) is $$not defined

Question 4

Which of the following functions is continuous for all real numbers?

Accepted Answer

$$f(x) = x^2$ + 3x - 5$$

Question 5

For which values of $$x is $$the function $$f(x) = \dfrac{x+1}{x-2}$$ continuous?

Accepted Answer

$$x 
eq 2$$

Question 6

Suppose $$f(x) is $$continuous on $$(1, 8)$$ and $$f(x) 
eq 0$$ for all $$x in$$ $$(1, 8). If$$ $$f(2) = 5$$, which of the following must be true?

Accepted Answer

$$f(x) is $$either always positive or always negative on $$(1, 8)$$

Question 7

Given $$f(x) = \sqrt{x+1}$$, for which values of $$x is$$ $$f(x)$$ continuous?

Accepted Answer

$$x \geq -1$$

Question 8

A bakery's profit function is $$P(x) = 6x - 0.001x^2$ - 5000. $$For which values of $$x is $$the profit positive?

Accepted Answer

$$1000 \u003c x \u003c 5000$$

Question 9

Which of the following is a partition number for the function $$f(x) = \dfrac{x+1}{x-2}$$?

Accepted Answer

$$x = 2$$

Question 10

If $$f(x) is $$continuous and nonzero on the interval $$(a, b)$$, what can be said about the sign of $$f(x) on$$ $$(a, b)$$?

Accepted Answer

$$f(x) is $$either always positive or always negative on $$(a, b)$$

Question 11

Which of the following statements about rational functions is true?

Accepted Answer

A rational function is continuous except where the denominator is zero

Question 12

Given the sign chart in the third image, for which intervals is $$f(x)$$ positive?

Accepted Answer

$$(-\infty, -1)$$ and $$(2, \infty)$$

Question 13

If $$f(x) is $$discontinuous at $$x = c$$, which of the following could be true?

Accepted Answer

$$f(c) is $$not defined

Question 14

Which of the following is NOT a property of continuous functions?

Accepted Answer

The quotient of two continuous functions is continuous everywhere

Question 15

Suppose $$f(x) is $$continuous everywhere and $$f(x) = 0 at$$ $$x = 3. $$What does this tell you about the sign of $$f(x)$$ near $$x = 3$$?

Accepted Answer

$$f(x)$$ changes sign at $$x = 3$$

Question 16

If $$f(x) is $$continuous at $$x = c$$ and $$f(c) is $$defined, which of the following must be true?

Accepted Answer

$$\displaystyle \lim_{x 	o c} f(x) = f(c)$$

Continuity and Sign Charts in Business Calculus 2.3

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