Q: For the function $$h(x) = \frac{|x|}{x}$$, which of the following statements is true?

$$\lim_{x \to 0^-} h(x) = -1$$ and $$\lim_{x \to 0^+} h(x) = 1$$

Q: Which property allows you to compute $$\lim_{x \to c} [k \cdot f(x)]$$ for a constant $$k$$?

$$\lim_{x \to c} [k \cdot f(x)] = k \cdot \lim_{x \to c} f(x)$$

Q: Which of the following is a correct algebraic property of limits?

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

Question 1

Which of the following best describes the concept of a limit in calculus?

Accepted Answer

The value that a function f(x) approaches as x approaches a specific point c, regardless of whether f(c) is defined.

Question 2

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

Accepted Answer

$$4$$

Question 3

For the function $$h(x) = \frac{|x|}{x}$$, which of the following statements is true?

Accepted Answer

$$\lim_{x 	o 0^-} h(x) = -1$$ and $$\lim_{x 	o 0^+} h(x) = 1$$

Question 4

If $$g(x) = \begin{cases} 1 + x & x 
eq 3 \ 3 & x = 3 \end{cases}$$, what is $$\lim_{x 	o 3} g(x)$$?

Accepted Answer

$$4$$

Question 5

Which statement about limits is correct?

Accepted Answer

A limit can exist even if the function is not defined at that point.

Question 6

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

Accepted Answer

$$5$$

Question 7

Which of the following is true about the limit $$\lim_{x 	o c} f(x)$$ for a polynomial $$f(x)$$?

Accepted Answer

It is always equal to $$f(c).$$

Question 8

Evaluate $$\lim_{x 	o 1} \sqrt{x^2$ + 3x + 1}.$$

Accepted Answer

$$\sqrt{5}$$

Question 9

If $$\lim_{x 	o c} f(x) = L$$ and $$\lim_{x 	o c} g(x) = M$$, what is $$\lim_{x 	o c} [f(x) + g(x)]$$?

Accepted Answer

$$L + M$$

Question 10

A rational function $$r(x) = \frac{f(x)}{g(x)}$$ has $$g(c) = 0. $$What can you conclude about $$\lim_{x 	o c} r(x)$$?

Accepted Answer

The limit does not exist unless simplification removes the zero denominator.

Question 11

Which property allows you to compute $$\lim_{x 	o c} [k \cdot f(x)]$$ for a constant $$k$$?

Accepted Answer

$$\lim_{x 	o c} [k \cdot f(x)] = k \cdot \lim_{x 	o c} f(x)$$

Question 12

If $$\lim_{x 	o c^-} f(x) = L$$ and $$\lim_{x 	o c^+} f(x) = M$$ with $$L 
eq M$$, what can you say about $$\lim_{x 	o c} f(x)$$?

Accepted Answer

The limit does not exist.

Question 13

A company models its profit as $$P(x) = 2x^2$ + 5x + 3. $$What is $$\lim_{x 	o 4} P(x)$$?

Accepted Answer

$$51$$

Question 14

Which of the following is a correct algebraic property of limits?

Accepted Answer

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

Question 15

Suppose $$f(x) = x^3$ - 2x + 7. $$What is $$\lim_{x 	o 2} f(x)$$?

Accepted Answer

$$15$$

Question 16

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

Accepted Answer

$$6$$

Question 17

Which of the following is true for the function $$f(x) = x$$?

Accepted Answer

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

Question 18

A rational function $$R(x) = \frac{x^2$ - 1}{x - 1} is $$used to model a chemical reaction rate, but the denominator is zero at $$x = 1. $$What is $$\lim_{x 	o 1} R(x)$$?

Accepted Answer

$$2$$

Question 19

If $$f(x) = 5$$ for all $$x$$, what is $$\lim_{x 	o 10} f(x)$$?

Accepted Answer

$$5$$

Limits: Foundations and Properties in Business Calculus

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