1.5.3: Additional Exercises

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Reading Questions

What does it mean for a function $f(x)$ to have a finite limit $L$ as $x$ approaches infinity?
How is a horizontal asymptote $y=L$ defined in terms of limits at infinity?
Can a function's graph cross its horizontal asymptote? If so, provide an example or describe one from the text.
State the Product Law for Limits at Infinity. What conditions must be met for it to apply?
What is the value of $ \displaystyle \lim_{x \to \infty} \frac{1}{x^p}$ if $p > 0$? Does this also hold for $x \to -\infty$? Are there any additional conditions for the $x \to -\infty$ case?
When evaluating the limit of a rational function at infinity, what is the common first algebraic step described in Example 2a?
In Example 2b, when simplifying $ \displaystyle \lim_{x \to \infty} \frac{\sqrt{1-x+x^2}}{3-x}$, how is $\sqrt{x^2(\frac{1}{x^2} - \frac{1}{x} + 1)}$ simplified when $x \to \infty$?
How does the simplification in the previous question change if $x \to -\infty$, as in Example 2c? What does $\sqrt{x^2}$ become?
What indeterminate form is encountered in Example 2d, $ \displaystyle \lim_{x \to -\infty} (\sqrt{1-x+9x^2} + 3x)$? What algebraic technique is used to resolve it?
What are the values of $ \displaystyle \lim_{x \to \infty} \tan^{-1}(x)$ and $ \displaystyle \lim_{x \to -\infty} \tan^{-1}(x)$?
State the Squeeze Theorem for Limits at Infinity.
What does it mean if $ \displaystyle \lim_{x \to \infty} f(x) = \infty$?
When evaluating an infinite limit at infinity for a rational function, such as $ \displaystyle \lim_{x \to -\infty} \frac{3+x^6}{x^4+8}$ in Example 1.5.5, what was the general strategy used after dividing by the highest power of $x$ from the original expression?

Homework

For exercises 1 - 5, examine the graphs. Identify where the vertical asymptotes are located and use limit notation in your answer.

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

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 using limits and without using graphing technology.

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) $\displaystyle \lim_{x \to \infty }\frac{1}{3x+6}$

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

12) $\displaystyle \lim_{x \to \infty }\frac{2x−5}{4x}$

13) $\displaystyle \lim_{x \to \infty }\frac{x^2−2x+5}{x+2}$

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

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

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

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

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

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

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

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

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

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

20) $\displaystyle \lim_{x \to \infty }\frac{2\sqrt{x}}{x−\sqrt{x}+1}$

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: $x= \pm 2$

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: $x= \pm 1$

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

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

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

41) $\displaystyle \lim_{x \to − \infty }x^2+10x+25$

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

42) $\displaystyle \lim_{x \to − \infty }\frac{x+2}{x^2+7x+6}$

43) $\displaystyle \lim_{x \to \infty }\frac{3x+2}{x+5}$

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

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