# 4.6E: Exercises for Section 4.6

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

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

1)

$$x=1$$

2)

3)

$$x=−1,\;x=2$$

4)

5)

$$x=0$$

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) $$f(x)=\dfrac{x}{x−2},\quad a=2$$

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

8) $$f(x)=(x+2)^{3/2},\quad a=−2$$

9) $$f(x)=(x−1)^{−1/3},\quad a=1$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

21) $$f(x)=x−\dfrac{9}{x}$$

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

22) $$f(x)=\dfrac{1}{1−x^2}$$

23) $$f(x)=\dfrac{x^3}{4−x^2}$$

Horizontal: none,
Vertical: $$x=±2$$

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

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

Horizontal: none,
Vertical: none

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

27) $$f(x)=\dfrac{x\sin(x)}{x^2−1}$$

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

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

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

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

30) $$f(x)=\dfrac{1}{x−1}−2x$$

31) $$f(x)=\dfrac{x^3+1}{x^3−1}$$

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

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

33) $$f(x)=x−\sin x$$

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 $$y=2$$

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

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

37) $$y=4, \;x=−1$$

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

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

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

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

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

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

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

$$\displaystyle \lim_{x→∞}\frac{3x+2}{x+5}=3$$ 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) $$y=x^3−3x^2+4$$

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

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

48) $$y=\dfrac{x^2+x−2}{x^2−3x−4}$$

49) $$y=\sqrt{x^2−5x+4}$$

50) $$y=2x\sqrt{16−x^2}$$

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

52) $$y=e^x−x^3$$

53) $$y=x\tan x, \quad x=[−π,π]$$

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

55) $$y=x^2\sin(x),\quad x=[−2π,2π]$$

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?

$$Q(x).$$ 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?

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