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

A value $$x = a$$ where $$\lim_{x \to a} f(x) = \infty or$$ $$-\infty.$$

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

Question 1

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

Accepted Answer

As $$x$$ approaches $$a$$, $$f(x)$$ becomes arbitrarily close to $$L.$$

Question 2

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

Accepted Answer

$$4$$

Question 3

If $$\lim_{x 	o 3^-} f(x) = 5$$ and $$\lim_{x 	o 3^+} f(x) = 7$$, what can you conclude about $$\lim_{x 	o 3} f(x)$$?

Accepted Answer

The limit does not exist.

Question 4

Evaluate $$\lim_{x 	o \infty} \frac{3x^2$ + 2}{5x^2$ - 7}.$$

Accepted Answer

$$\frac{3}{5}$$

Question 5

Which of the following is a correct application of the limit sum law?

Accepted Answer

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

Question 6

Suppose $$\lim_{x 	o 1} f(x) = 2$$ and $$\lim_{x 	o 1} g(x) = -3. $$What is $$\lim_{x 	o 1} [2f(x) - 3g(x)]$$?

Accepted Answer

$$2 \cdot 2 - 3 \cdot (-3) = 4 + 9 = 13$$

Question 7

A function $$f(x) is $$defined as $$f(x) = \frac{x^2$ - 9}{x - 3}. $$What is the value of $$\lim_{x 	o 3} f(x)$$?

Accepted Answer

$$6$$

Question 8

Which of the following best describes a vertical asymptote of a rational function $$f(x)$$?

Accepted Answer

A value $$x = a$$ where $$\lim_{x 	o a} f(x) = \infty or$$ $$-\infty.$$

Question 9

Evaluate $$\lim_{x 	o 0} \frac{\sin x}{x}.$$

Accepted Answer

$$1$$

Question 10

If $$\lim_{x 	o 2} f(x) = 5$$ and $$\lim_{x 	o 2} g(x) = -1$$, what is $$\lim_{x 	o 2} [f(x)g(x)]$$?

Accepted Answer

$$-5$$

Question 11

Given $$f(x) = \frac{2x^2$ + 3x - 2}{x^2$ - 1}$$, what is the horizontal asymptote as $$x 	o \infty$$?

Accepted Answer

$$y = 2$$

Question 12

Which of the following limits does NOT exist?

Accepted Answer

$$\lim_{x 	o 0} \frac{1}{x}$$

Question 13

Evaluate $$\lim_{x 	o 1} \frac{x^2$ - 1}{x - 1}.$$

Accepted Answer

$$2$$

Question 14

Suppose $$\lim_{x 	o 4} f(x) = 7$$ and $$\lim_{x 	o 4} g(x) = 2. $$What is $$\lim_{x 	o 4} \frac{f(x)}{g(x)}$$?

Accepted Answer

$$\frac{7}{2}$$

Question 15

Which of the following is the correct limit law for the product of two functions?

Accepted Answer

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

Question 16

Evaluate $$\lim_{x 	o 0} \frac{e^x - 1}{x}.$$

Accepted Answer

$$1$$

Question 17

A company’s profit function is $$P(x) = \frac{100x}{x+10}. $$What is the limiting profit as $$x 	o \infty$$?

Accepted Answer

$$100$$

Question 18

Which of the following is true about the limit $$\lim_{x 	o a} f(x) if$$ $$f(x)$$ has a hole at $$x = a$$?

Accepted Answer

The limit may still exist even if $$f(a) is $$undefined.

Question 19

Evaluate $$\lim_{x 	o 2} \frac{x^2$ - 4}{x^2$ + 2x - 8}.$$

Accepted Answer

$$0$$

Limits and Their Applications in Business Calculus

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