Q: Which of the following is a correct application of the Sum Law for limits?

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

Q: Which of the following statements about vertical asymptotes is correct?

If $$\lim_{x \to a} f(x) = \pm \infty$$, then $$x = a is a $$vertical asymptote of $$f(x).$$

Q: Evaluate $$\lim_{x \to \infty} (3x^4$ - 6x^2$ + x - 10). $$What does this tell you about the end behavior of the function?

$$+\infty$$; the function increases without bound as $$x \to \infty$$

Q: Which of the following is the correct definition of a horizontal asymptote?

If $$\lim_{x \to \infty} f(x) = L$$, then $$y = L is a $$horizontal asymptote.

Q: Evaluate $$\lim_{x \to -\infty} (-2x^3$ + 3x^2$ - 12). $$What does this tell you about the end behavior?

$$+\infty$$; the function increases without bound as $$x \to -\infty$$

Q: Which of the following is a correct application of the Quotient Law for limits?

$$\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$$, \text{provided } \lim_{x \to a} g(x) \neq 0$

Question 1

Suppose $$s(t)$$ represents the position (in meters) of a car at time $$t (in $$seconds). If $$s(5) = 120$$ and $$s(2) = 60$$, what is the average velocity of the car between $$t = 2$$ and $$t = 5$$?

Accepted Answer

$$\frac{120 - 60}{5 - 2} = 20 m/s$$

Question 2

Evaluate $$\lim_{x 	o 4} \frac{x^2$ - 16}{x - 4}$$ using limit laws.

Accepted Answer

$$8$$

Question 3

Which of the following is a correct application of the Sum Law for limits?

Accepted Answer

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

Question 4

Given $$f(x) = \frac{1}{x(x+1)}$$, what is $$\lim_{x 	o -1^-} f(x)$$?

Accepted Answer

$$+\infty$$

Question 5

If $$\lim_{x 	o 3} f(x) = -75$$ and $$\lim_{x 	o 3} g(x) = 0$$ with $$g(x) \u003e 0 as$$ $$x 	o 3$$, what is $$\lim_{x 	o 3} \frac{f(x)}{g(x)}$$?

Accepted Answer

$$-\infty$$

Question 6

Which of the following statements about vertical asymptotes is correct?

Accepted Answer

If $$\lim_{x 	o a} f(x) = \pm \infty$$, then $$x = a is a $$vertical asymptote of $$f(x).$$

Question 7

Evaluate $$\lim_{x 	o \infty} (3x^4$ - 6x^2$ + x - 10). $$What does this tell you about the end behavior of the function?

Accepted Answer

$$+\infty$$; the function increases without bound as $$x 	o \infty$$

Question 8

Which of the following is the correct definition of a horizontal asymptote?

Accepted Answer

If $$\lim_{x 	o \infty} f(x) = L$$, then $$y = L is a $$horizontal asymptote.

Question 9

Find all vertical asymptotes of $$f(x) = \frac{2x}{x^2$ - 9}.$$

Accepted Answer

$$x = 3$$ and $$x = -3$$

Question 10

Determine $$\lim_{x 	o 0^+} \frac{1}{x}.$$

Accepted Answer

$$+\infty$$

Question 11

Which of the following is true about the limit $$\lim_{x 	o \infty} \frac{5x^2$ + 3}{2x^2$ - x}$$?

Accepted Answer

$$\frac{5}{2}$$

Question 12

If $$f(x) is $$continuous at $$x = 2$$, which of the following must be true?

Accepted Answer

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

Question 13

Which of the following functions is discontinuous at $$x = 0$$?

Accepted Answer

$$f(x) = \frac{1}{x}$$

Question 14

Evaluate $$\lim_{x 	o -\infty} (-2x^3$ + 3x^2$ - 12). $$What does this tell you about the end behavior?

Accepted Answer

$$+\infty$$; the function increases without bound as $$x 	o -\infty$$

Question 15

Which limit law allows you to compute $$\lim_{x 	o a} [c \cdot f(x)] = c \cdot \lim_{x 	o a} f(x)$$?

Accepted Answer

Constant Multiple Law

Question 16

Suppose $$\lim_{x 	o 2} f(x) = 5$$ and $$\lim_{x 	o 2} g(x) = -3. $$What is $$\lim_{x 	o 2} [f(x)g(x)]$$?

Accepted Answer

$$-15$$

Question 17

Which of the following is true about $$\lim_{x 	o \infty} x^n$$ for even $$n$$?

Accepted Answer

$$\infty$$

Question 18

If $$f(x) = \sqrt{x}$$, what is $$\lim_{x 	o 0^+} f(x)$$?

Accepted Answer

$$0$$

Question 19

Which of the following is a correct application of the Quotient Law for limits?

Accepted Answer

$$\lim_{x 	o a} \frac{f(x)}{g(x)} = \frac{\lim_{x 	o a} f(x)}{\lim_{x 	o a} g(x)}$$, 	ext{provided } \lim_{x 	o a} g(x) 
eq 0$

Calculus I Exam 1 Review: Limits, Continuity, and Asymptotes

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