Q: Evaluate $$\lim_{x \to 4} \frac{x^2$ - 16}{x - 4}$$ and state the limit law(s) used.

$$\lim_{x \to 4} \frac{x^2$ - 16}{x - 4} = 8$$; used the quotient, sum, and power laws.

Q: Which of the following is the 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 horizontal asymptotes is correct?

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

Q: Which of the following is NOT a valid limit law?

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

Q: Which of the following describes the behavior of $$f(x) = -2x^3$ + 3x^2$ - 12 as$$ $$x \to -\infty$$?

$$\lim_{x \to -\infty} f(x) = +\infty$$

Question 1

Suppose a car travels along a straight road and its position at time $$t is $$given by $$s(t) = 3t^2$ + 2t. $$What is the average velocity of the car from $$t = 1 to$$ $$t = 4$$?

Accepted Answer

$$v_{av} = \frac{s(4) - s(1)}{4 - 1} = \frac{54 - 5}{3} = 16.33$$

Question 2

Evaluate $$\lim_{x 	o 4} \frac{x^2$ - 16}{x - 4}$$ and state the limit law(s) used.

Accepted Answer

$$\lim_{x 	o 4} \frac{x^2$ - 16}{x - 4} = 8$$; used the quotient, sum, and power laws.

Question 3

Which of the following is the 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

Evaluate $$\lim_{x 	o -2} (x^3$ - 3x^2$ + 4x - 5).$$

Accepted Answer

$$-33$$

Question 5

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

Accepted Answer

$$+\infty$$

Question 6

Suppose $$\lim_{x 	o 3} f(x) = -75$$ 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 7

Which of the following functions has a vertical asymptote at $$x = 2$$?

Accepted Answer

$$f(x) = \frac{1}{x-2}$$

Question 8

Evaluate $$\lim_{x 	o \infty} (3x^4$ - 6x^2$ + x - 10). $$What theorem justifies your answer?

Accepted Answer

$$+\infty$$; by Theorem 2.6, the highest degree term dominates.

Question 9

Which of the following statements about horizontal asymptotes is correct?

Accepted Answer

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

Question 10

Find all points of discontinuity for $$f(x) = \frac{x^2$ - 9}{x^2$ - 4}.$$

Accepted Answer

$$x = 2$$ and $$x = -2$$

Question 11

Which of the following is NOT a valid limit law?

Accepted Answer

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

Question 12

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

Accepted Answer

$$+\infty$$

Question 13

Which of the following describes the behavior of $$f(x) = -2x^3$ + 3x^2$ - 12 as$$ $$x 	o -\infty$$?

Accepted Answer

$$\lim_{x 	o -\infty} f(x) = +\infty$$

Question 14

If $$\lim_{x 	o a} f(x) = \infty$$, what can you conclude about the graph of $$f(x)$$?

Accepted Answer

There is a vertical asymptote at $$x = a.$$

Question 15

Evaluate $$\lim_{x 	o 0} \sqrt{x^2$}$$ using the root law.

Accepted Answer

$$0$$

Question 16

Which of the following is true about the continuity of $$f(x) = \frac{1}{x} at$$ $$x = 0$$?

Accepted Answer

$$f(x) is $$not continuous at $$x = 0$$ because the limit does not exist.

Question 17

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

Accepted Answer

$$\infty$$

Question 18

Which of the following is a correct statement about limits at infinity for polynomials?

Accepted Answer

The term with the highest degree determines the limit as $$x 	o \infty.$$

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

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