4.6E: Exercises for Section 4.6

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

1)

Answer

$x=1$

2)

3)

Answer

$x=−1,\;x=2$

4)

5)

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 $y=3$.

In exercises 44 - 55, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior.

44) $y=3x^2+2x+4$

45) $y=x^3−3x^2+4$

Answer

46) $y=\dfrac{2x+1}{x^2+6x+5}$

47) $y=\dfrac{x^3+4x^2+3x}{3x+9}$

Answer

48) $y=\dfrac{x^2+x−2}{x^2−3x−4}$

49) $y=\sqrt{x^2−5x+4}$

Answer

50) $y=2x\sqrt{16−x^2}$

51) $y=\dfrac{\cos x}{x}$, on $x=[−2π,2π]$

Answer

52) $y=e^x−x^3$

53) $y=x\tan x, \quad x=[−π,π]$

Answer

54) $y=x\ln(x), \quad x>0$

55) $y=x^2\sin(x),\quad x=[−2π,2π]$

Answer

56) For $f(x)=\dfrac{P(x)}{Q(x)}$ to have an asymptote at $y=2$ then the polynomials $P(x)$ and $Q(x)$ must have what relation?

57) For $f(x)=\dfrac{P(x)}{Q(x)}$ to have an asymptote at $x=0$, then the polynomials $P(x)$ and $Q(x).$ must have what relation?

Answer

$Q(x).$ must have have $x^{k+1}$ as a factor, where $P(x)$ has $x^k$ as a factor.

58) If $f′(x)$ has asymptotes at $y=3$ and $x=1$, then $f(x)$ has what asymptotes?

59) Both $f(x)=\dfrac{1}{x−1}$ and $g(x)=\dfrac{1}{(x−1)^2}$ have asymptotes at $x=1$ and $y=0.$ What is the most obvious difference between these two functions?

Answer

$\displaystyle \lim_{x→1^−}f(x)=-\infty \text{ and } \lim_{x→1^−}g(x)=\infty$

60) True or false: Every ratio of polynomials has vertical asymptotes.