Limits at Infinity

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Show your work for each problem.

1.
Find $\lim_{x \to \infty} \frac{7}{x^3}$ .
1. a)no finite limit
2. b) $3$
3. c) $7$
4. d) $0$
2.
Find $\lim_{x \to \infty} \frac{5x^2 + 3}{x^2 - 4}$ .
3.
Find $\lim_{x \to \infty} \left(4 + \frac{1}{x}\right)$ .
4.
What is the horizontal asymptote of $y = \frac{8x + 1}{2x - 3}$ ?
1. a) $y = 0$
2. b) $y = 4$
3. c) $y = 8$
4. d) $y = -3$
5.
Find $\lim_{x \to \infty} \left(6 - 2e^{-x}\right)$ .
6.
Find $\lim_{x \to -\infty} \frac{2x + 1}{x - 5}$ .
7.
Find $\lim_{x \to \infty} \frac{3x^2 - x}{5x^2 + 2}$ . Write the answer as a fraction.
8.
What is $\lim_{x \to \infty} \frac{x^4 + 1}{x^3 - 2x}$ ?
1. a) $0$
2. b)no finite limit, the values grow without bound
3. c) $4$
4. d) $1$
9.
Find $\lim_{x \to \infty} \frac{10x^2 + 3x}{5x^2 - 7}$ by dividing by the highest power of $x$ .
10.
Find $\lim_{x \to \infty} \frac{4x + 9}{x^2 + 1}$ .
11.
The graph of $f(x) = \frac{12x^3 - 5}{4x^3 + x^2}$ has the horizontal asymptote $y = L$ . Find $L$ .
12.
Which function has the horizontal asymptote $y = 0$ ?
1. a) $y = 5 + e^{-x}$
2. b) $y = \frac{2x^2 + 1}{x^2}$
3. c) $y = \frac{x^3}{x + 1}$
4. d) $y = \frac{x + 4}{x^2 - 9}$
13.
Find $\lim_{x \to -\infty} \frac{3 - 6x^3}{2x^3 + x}$ .
14.
Find $\lim_{x \to \infty} \frac{(2x + 3)^2}{x^2 + 1}$ .
15.
What happens to $\frac{5x^2 + 1}{3 - 2x}$ as $x \to -\infty$ ?
1. a)it decreases without bound ( $-\infty$ )
2. b) $0$
3. c) $-\frac{5}{2}$
4. d)it grows without bound ( $+\infty$ )
16.
A company's average cost per item is $C(x) = \frac{2000 + 8x}{x}$ dollars for $x$ items. What does the average cost approach as $x$ grows without bound?
17.
For which value of $k$ does the graph of $y = \frac{kx^2 + 1}{4x^2 - 3}$ have the horizontal asymptote $y = 5$ ?
18.
Find $\lim_{x \to \infty} \frac{6 + 2e^{-x}}{3 - e^{-x}}$ .