2.5E: Exercises for Section 2.5

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For exercises 1 - 5, examine the graphs. Identify where the vertical asymptotes are located.

$The function graphed decreases very rapidly as it approaches x = 1 from the left, and on the other side of x = 1, it seems to start near infinity and then decrease rapidly.$

Answer: $x=1$

$The function graphed increases very rapidly as it approaches x = −3 from the left, and on the other side of x = −3, it seems to start near negative infinity and then increase rapidly to form a sort of U shape that is pointing down, with the other side of the U being at x = 2. On the other side of x = 2, the graph seems to start near infinity and then decrease rapidly.$

$The function graphed decreases very rapidly as it approaches x = −1 from the left, and on the other side of x = −1, it seems to start near negative infinity and then increase rapidly to form a sort of U shape that is pointing down, with the other side of the U being at x = 2. On the other side of x = 2, the graph seems to start near infinity and then decrease rapidly.$

Answer: $x=−1,\;x=2$

$The function graphed decreases very rapidly as it approaches x = 0 from the left, and on the other side of x = 0, it seems to start near infinity and then decrease rapidly to form a sort of U shape that is pointing up, with the other side of the U being at x = 1. On the other side of x = 1, there is another U shape pointing down, with its other side being at x = 2. On the other side of x = 2, the graph seems to start near negative infinity and then increase rapidly.$

$The function graphed decreases very rapidly as it approaches x = 0 from the left, and on the other side of x = 0, it seems to start near infinity and then decrease rapidly to form a sort of U shape that is pointing up, with the other side being a normal function that appears as if it will take the entirety of the values of the x-axis.$

Answer: $x=0$

For the functions $f(x)$ in exercises 6 - 10, determine whether there is an asymptote at $x=a$. Justify your answer without graphing on a calculator.

6) $f(x)=\dfrac{x+1}{x^2+5x+4},\quad a=−1$

7) $f(x)=\dfrac{x}{x−2},\quad a=2$

Answer: Yes, there is a vertical asymptote at $x = 2$.

8) $f(x)=(x+2)^{3/2},\quad a=−2$

9) $f(x)=(x−1)^{−1/3},\quad a=1$

Answer: Yes, there is vertical asymptote at $x = 1$.

10) $f(x)=1+x^{−2/5},\quad a=1$

In exercises 11 - 20, evaluate the limit.

11) $\displaystyle \lim_{x→∞}\frac{1}{3x+6}$

Answer: $\displaystyle \lim_{x→∞}\frac{1}{3x+6} = 0$

12) $\displaystyle \lim_{x→∞}\frac{2x−5}{4x}$

13) $\displaystyle \lim_{x→∞}\frac{x^2−2x+5}{x+2}$

Answer: $\displaystyle \lim_{x→∞}\frac{x^2−2x+5}{x+2} = ∞$

14) $\displaystyle \lim_{x→−∞}\frac{3x^3−2x}{x^2+2x+8}$

15) $\displaystyle \lim_{x→−∞}\frac{x^4−4x^3+1}{2−2x^2−7x^4}$

Answer: $\displaystyle \lim_{x→−∞}\frac{x^4−4x^3+1}{2−2x^2−7x^4} = −\frac{1}{7}$

16) $\displaystyle \lim_{x→∞}\frac{3x}{\sqrt{x^2+1}}$

17) $\displaystyle \lim_{x→−∞}\frac{\sqrt{4x^2−1}}{x+2}$

Answer: $\displaystyle \lim_{x→−∞}\frac{\sqrt{4x^2−1}}{x+2} = -2$

18) $\displaystyle \lim_{x→∞}\frac{4x}{\sqrt{x^2−1}}$

19) $\displaystyle \lim_{x→−∞}\frac{4x}{\sqrt{x^2−1}}$

Answer: $\displaystyle \lim_{x→−∞}\frac{4x}{\sqrt{x^2−1}} = -4$

20) $\displaystyle \lim_{x→∞}\frac{2\sqrt{x}}{x−\sqrt{x}+1}$

For exercises 21 - 25, find the horizontal and vertical asymptotes.

21) $f(x)=x−\dfrac{9}{x}$

Answer: Horizontal: none,
Vertical: $x=0$

22) $f(x)=\dfrac{1}{1−x^2}$

23) $f(x)=\dfrac{x^3}{4−x^2}$

Answer: Horizontal: none,
Vertical: $x=±2$

24) $f(x)=\dfrac{x^2+3}{x^2+1}$

25) $f(x)=\sin(x)\sin(2x)$

Answer: Horizontal: none,
Vertical: none

26) $f(x)=\cos x+\cos(3x)+\cos(5x)$

27) $f(x)=\dfrac{x\sin(x)}{x^2−1}$

Answer: Horizontal: $y=0,$
Vertical: $x=±1$

28) $f(x)=\dfrac{x}{\sin(x)}$

29) $f(x)=\dfrac{1}{x^3+x^2}$

Answer: Horizontal: $y=0,$
Vertical: $x=0$ and $x=−1$

30) $f(x)=\dfrac{1}{x−1}−2x$

31) $f(x)=\dfrac{x^3+1}{x^3−1}$

Answer: Horizontal: $y=1,$
Vertical: $x=1$

32) $f(x)=\dfrac{\sin x+\cos x}{\sin x−\cos x}$

33) $f(x)=x−\sin x$

Answer: Horizontal: none,
Vertical: none

34) $f(x)=\dfrac{1}{x}−\sqrt{x}$

For exercises 35 - 38, construct a function $f(x)$ that has the given asymptotes.

35) $x=1$ and $y=2$

Answer: Answers will vary, for example: $y=\dfrac{2x}{x−1}$

36) $x=1$ and $y=0$

37) $y=4, \;x=−1$

Answer: Answers will vary, for example: $y=\dfrac{4x}{x+1}$

38) $x=0$

In exercises 39 - 43, graph the function on a graphing calculator on the window $x=[−5,5]$ and estimate the horizontal asymptote or limit. Then, calculate the actual horizontal asymptote or limit.

39) [T] $f(x)=\dfrac{1}{x+10}$

Answer: $\displaystyle \lim_{x→∞}\frac{1}{x+10}=0$ so $f$ has a horizontal asymptote of $y=0$.

40) [T] $f(x)=\dfrac{x+1}{x^2+7x+6}$

41) [T] $\displaystyle \lim_{x→−∞}x^2+10x+25$

Answer: $\displaystyle \lim_{x→−∞}x^2+10x+25 = ∞$

42) [T] $\displaystyle \lim_{x→−∞}\frac{x+2}{x^2+7x+6}$

43) [T] $\displaystyle \lim_{x→∞}\frac{3x+2}{x+5}$

Answer: $\displaystyle \lim_{x→∞}\frac{3x+2}{x+5}=3$ so this function has a horizontal asymptote of $y=3$.

Contributors and Attributions

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

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