Limits at Infinity

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) $0$
Answer: $0$
As $x$ grows without bound, $x^3$ grows without bound too, so $\frac{7}{x^3}$ gets closer and closer to $0$ . The limit is $0$ .
2.
Find $\lim_{x \to \infty} \frac{5x^2 + 3}{x^2 - 4}$ .
Answer: 5
Both parts have degree $2$ , so the limit is the ratio of the leading coefficients: $\frac{5}{1} = 5$ .
3.
Find $\lim_{x \to \infty} \left(4 + \frac{1}{x}\right)$ .
Answer: 4
1. What is $\frac{1}{x}$ at $x = 1000$ ? Write it as a decimal. 0.001
2. What does $\frac{1}{x}$ approach as $x \to \infty$ ? 0
3. What is $\lim_{x \to \infty} \left(4 + \frac{1}{x}\right)$ ? 4
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) $y = -3$
Answer: $y = 4$
The degrees are equal, so the asymptote is at the ratio of the leading coefficients: $y = \frac{8}{2} = 4$ .
5.
Find $\lim_{x \to \infty} \left(6 - 2e^{-x}\right)$ .
Answer: 6
$e^{-x} \to 0$ , so $2e^{-x} \to 0$ and $6 - 2e^{-x} \to 6 - 0 = 6$ .
6.
Find $\lim_{x \to -\infty} \frac{2x + 1}{x - 5}$ .
Answer: 2
1. What is the degree of the numerator? 1
2. What is the degree of the denominator? 1
3. The degrees are equal. What is the limit $\frac{2}{1}$ ? 2
7.
Find $\lim_{x \to \infty} \frac{3x^2 - x}{5x^2 + 2}$ . Write the answer as a fraction.
Answer: 3/5
Dividing by $x^2$ gives $\frac{3 - \frac{1}{x}}{5 + \frac{2}{x^2}} \to \frac{3}{5}$ .
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) $1$
Answer: no finite limit, the values grow without bound
The numerator has degree $4$ and the denominator degree $3$ . Dividing by $x^3$ gives $\frac{x + \frac{1}{x^3}}{1 - \frac{2}{x^2}}$ , which grows like $x$ , so there is no finite limit.
9.
Find $\lim_{x \to \infty} \frac{10x^2 + 3x}{5x^2 - 7}$ by dividing by the highest power of $x$ .
Answer: 2
1. The highest power of $x$ in the denominator is $x^n$ . What is $n$ ? 2
2. After dividing by $x^2$ the numerator is $10 + \frac{3}{x}$ . What does it approach? 10
3. The denominator is now $5 - \frac{7}{x^2}$ . What does it approach? 5
4. What is the limit? 2
10.
Find $\lim_{x \to \infty} \frac{4x + 9}{x^2 + 1}$ .
Answer: 0
Degree $1$ on top, degree $2$ below. Dividing by $x^2$ gives $\frac{\frac{4}{x} + \frac{9}{x^2}}{1 + \frac{1}{x^2}} \to \frac{0}{1} = 0$ .
11.
The graph of $f(x) = \frac{12x^3 - 5}{4x^3 + x^2}$ has the horizontal asymptote $y = L$ . Find $L$ .
Answer: 3
1. What is the degree of the numerator? 3
2. What is the degree of the denominator? 3
3. What is the leading coefficient of the numerator? 12
4. What is the leading coefficient of the denominator? 4
5. What is $L$ ? 3
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) $y = \frac{x + 4}{x^2 - 9}$
Answer: $y = \frac{x + 4}{x^2 - 9}$
In $\frac{x + 4}{x^2 - 9}$ the numerator has degree $1$ and the denominator degree $2$ , so the limit is $0$ . The others approach $2$ , grow without bound, or approach $5$ .
13.
Find $\lim_{x \to -\infty} \frac{3 - 6x^3}{2x^3 + x}$ .
Answer: -3
1. What is the leading coefficient of the numerator $3 - 6x^3$ ? -6
2. What is the leading coefficient of the denominator $2x^3 + x$ ? 2
3. Both degrees are 3. What is the limit? -3
14.
Find $\lim_{x \to \infty} \frac{(2x + 3)^2}{x^2 + 1}$ .
Answer: 4
1. Expanding gives $(2x + 3)^2 = ax^2 + 12x + 9$ . What is $a$ ? 4
2. What is the degree of the numerator? 2
3. The degrees are equal. What is the limit? 4
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 ( $+\infty$ )
Answer: it grows without bound ( $+\infty$ )
Degree $2$ on top, degree $1$ below: no finite limit. For very negative $x$ the numerator is positive and $3 - 2x$ is positive too, so the fraction grows without bound: at $x = -1000$ it is about $4985$ .
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?
Answer: 8
1. What is $C(500)$ ? 12
2. Write $C(x) = \frac{2000}{x} + 8$ . What does $\frac{2000}{x}$ approach? 0
3. What is the limit of $C(x)$ ? 8
17.
For which value of $k$ does the graph of $y = \frac{kx^2 + 1}{4x^2 - 3}$ have the horizontal asymptote $y = 5$ ?
Answer: 20
Equal degrees, so the asymptote is $y = \frac{k}{4}$ . From $\frac{k}{4} = 5$ we get $k = 20$ .
18.
Find $\lim_{x \to \infty} \frac{6 + 2e^{-x}}{3 - e^{-x}}$ .
Answer: 2
1. What does $e^{-x}$ approach as $x \to \infty$ ? 0
2. What does the numerator $6 + 2e^{-x}$ approach? 6
3. What does the denominator $3 - e^{-x}$ approach? 3
4. What is the limit? 2