Problem Set: Limits at Infinity and Asymptotes

For the following exercises, 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.

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$x=1$

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.

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$x=-1, \, x=2$

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.

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$x=0$

For the following functions $f(x)$ , determine whether there is an asymptote at $x=a$ . Justify your answer without graphing on a calculator.

6. $f(x)=\dfrac{x+1}{x^2+5x+4}, \, a=-1$

7. $f(x)=\dfrac{x}{x-2}, \, a=2$

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8. $f(x)=(x+2)^{3/2}, \, a=-2$

9. $f(x)=(x-1)^{-1/3}, \, a=1$

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10. $f(x)=1+x^{-2/5}, \, a=1$

For the following exercises, evaluate the limit.

11. $\underset{x\to \infty }{\lim}\dfrac{1}{3x+6}$

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12. $\underset{x\to \infty }{\lim}\dfrac{2x-5}{4x}$

13. $\underset{x\to \infty }{\lim}\dfrac{x^2-2x+5}{x+2}$

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$\infty$

14. $\underset{x\to −\infty }{\lim}\dfrac{3x^3-2x}{x^2+2x+8}$

15. $\underset{x\to −\infty }{\lim}\dfrac{x^4-4x^3+1}{2-2x^2-7x^4}$

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$-\frac{1}{7}$

16. $\underset{x\to \infty }{\lim}\dfrac{3x}{\sqrt{x^2+1}}$

17. $\underset{x\to −\infty }{\lim}\dfrac{\sqrt{4x^2-1}}{x+2}$

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18. $\underset{x\to \infty }{\lim}\dfrac{4x}{\sqrt{x^2-1}}$

19. $\underset{x\to −\infty }{\lim}\dfrac{4x}{\sqrt{x^2-1}}$

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20. $\underset{x\to \infty }{\lim}\dfrac{2\sqrt{x}}{x-\sqrt{x}+1}$

For the following exercises, find the horizontal and vertical asymptotes.

21. $f(x)=x-\dfrac{9}{x}$

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Horizontal: none, vertical: $x=0$

22. $f(x)=\dfrac{1}{1-x^2}$

23. $f(x)=\dfrac{x^3}{4-x^2}$

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Horizontal: none, vertical: $x=\pm 2$

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

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

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

27. $f(x)=\dfrac{x \sin (x)}{x^2-1}$

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Horizontal: $y=0$ , vertical: $x=\pm 1$

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

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

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Horizontal: $y=0$ , vertical: $x=0$ and $x=-1$

30. $f(x)=\dfrac{1}{x-1}-2x$

31. $f(x)=\dfrac{x^3+1}{x^3-1}$

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

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

33. $f(x)=x- \sin x$

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34. $f(x)=\dfrac{1}{x}-\sqrt{x}$

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

35. $x=1$ and $y=2$

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

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

37. $y=4$ and $x=-1$

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

38. $x=0$

For the following exercises (40-44), 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] $f(x)=\dfrac{1}{x+10}$

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$y=0$

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

41. [T] $\underset{x\to −\infty }{\lim} x^2+10x+25$

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$\infty$

42. [T] $\underset{x\to −\infty }{\lim}\dfrac{x+2}{x^2+7x+6}$

43. [T] $\underset{x\to \infty }{\lim}\dfrac{3x+2}{x+5}$

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$y=3$

For the following exercises (45-56), 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. $y=x^3-3x^2+4$

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

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

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48. $y=\dfrac{x^2+x-2}{x^2-3x-4}$

49. $y=\sqrt{x^2-5x+4}$

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50. $y=2x\sqrt{16-x^2}$

51. $y=\dfrac{ \cos x}{x}$ , on $x=[-2\pi ,2\pi]$

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52. $y=e^x-x^3$

53. $y=x \tan x, \, x=[−\pi ,\pi]$

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

55. $y=x^2 \sin (x), \, x=[-2\pi ,2\pi]$

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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?

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The degree of $Q(x)$ must be greater than the degree of $P(x)$ .

$P(0)\ne{0}$ and $Q(x)=0$

58. If $f^{\prime}(x)$ 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?

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$\underset{x\to 1^-}{\lim} f(x)=-\infty$ and $\underset{x\to 1^-}{\lim} g(x)=\infty$

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

Applications of Derivatives: Supplemental Content

Problem Set: Limits at Infinity and Asymptotes

For the following exercises, evaluate the limit.

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