3.9E: Exercises for Section 3.9

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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 - 29, 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) $f(x)=\dfrac{3x^2-3x}{x^2+x-12}$

Answer: Horizontal: $y=3$
Vertical: $x=-4,x=3$

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

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

27) $f(x)=\dfrac{1}{x−1}−2x$

28) $f(x)=\dfrac{x^3+1}{x^3−1}$

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

29) $f(x)=\dfrac{1}{x}−\sqrt{x}$

For exercises 30-33, construct a function $f(x)$ that has the given asymptotes.

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

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

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

32) $y=4, \;x=−1$

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

33) $x=0$

In exercises 34-38 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.

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

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

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

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

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

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

38) [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 39-47, 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.

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

40) $y=x^3−3x^2+4$

Answer: $The function starts in the third quadrant, increases to pass through (−1, 0), increases to a maximum and y intercept at 4, decreases to touch (2, 0), and then increases to (4, 20).$

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

42) $y=\dfrac{2x+2}{x-1}$

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

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

Answer: $An upward-facing parabola with minimum between x = 0 and x = −1 with y intercept between 0 and 1.$

44) $y=\dfrac{x^2+x−2}{x^2−3x−4}$

45) $y=\sqrt{x^2−5x+4}$

Answer: $This graph starts at (−2, 4) and decreases in a convex way to (1, 0). Then the graph starts again at (4, 0) and increases in a convex way to (6, 3).$

46) $y=e^x−x^3$

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

48) 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?

49) 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.

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

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

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

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