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.