Stewart Section 2.6: Limits at Infinity; Horizontal Asymptotes

1 Limits at Infinity - Concepts · Level 1

Explain in your own words the meaning of each of the following.

(a) \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x) = 5\)

Answer for 1(a)

(b) \(\operatorname*{lim}\limits_{x \rightarrow -\infty} f(x) = 3\)

Answer for 1(b)

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2 Limits at Infinity - Concepts · Level 2

(a) Can the graph of \(y = f(x)\) intersect a vertical asymptote? Can it intersect a horizontal asymptote? Illustrate by sketching graphs.

Answer for 2(a)

(b) How many horizontal asymptotes can the graph of \(y = f(x)\) have? Sketch graphs to illustrate the possibilities.

Answer for 2(b)

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3 Limits at Infinity - Graph Analysis · Level 2

For the function \(f\) whose graph is given, state the following.

(a) \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x)\)

Answer for 3(a)

(b) \(\operatorname*{lim}\limits_{x \rightarrow -\infty} f(x)\)

Answer for 3(b)

(c) \(\operatorname*{lim}\limits_{x \rightarrow 1} f(x)\)

Answer for 3(c)

(d) \(\operatorname*{lim}\limits_{x \rightarrow 3} f(x)\)

Answer for 3(d)

(e) The equations of the asymptotes

Answer for 3(e)

Enter your answer directly below each part above.

4 Limits at Infinity - Graph Analysis · Level 2

For the function \(g\) whose graph is given, state the following.

(a) \(\operatorname*{lim}\limits_{x \rightarrow \infty} g(x)\)

Answer for 4(a)

(b) \(\operatorname*{lim}\limits_{x \rightarrow -\infty} g(x)\)

Answer for 4(b)

(c) \(\operatorname*{lim}\limits_{x \rightarrow 0} g(x)\)

Answer for 4(c)

(d) \(\operatorname*{lim}\limits_{x \rightarrow 2^-} g(x)\)

Answer for 4(d)

(e) \(\operatorname*{lim}\limits_{x \rightarrow 2^+} g(x)\)

Answer for 4(e)

(f) The equations of the asymptotes

Answer for 4(f)

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5 Limits at Infinity - Sketching · Level 3

Sketch the graph of an example of a function \(f\) that satisfies all of the given conditions. \(\operatorname*{lim}\limits_{x \rightarrow 0} f(x) = -\infty\), \(\operatorname*{lim}\limits_{x \rightarrow -\infty} f(x) = 5\), \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x) = -5\)

6 Limits at Infinity - Sketching · Level 3

Sketch the graph of an example of a function \(f\) that satisfies all of the given conditions. \(\operatorname*{lim}\limits_{x \rightarrow 2} f(x) = \infty\), \(\operatorname*{lim}\limits_{x \rightarrow -2^+} f(x) = \infty\), \(\operatorname*{lim}\limits_{x \rightarrow -2^-} f(x) = -\infty\), \(\operatorname*{lim}\limits_{x \rightarrow -\infty} f(x) = 0\), \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x) = 0\)

7 Limits at Infinity - Sketching · Level 3

Sketch the graph of an example of a function \(f\) that satisfies all of the given conditions. \(\operatorname*{lim}\limits_{x \rightarrow 2} f(x) = -\infty\), \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x) = \infty\), \(\operatorname*{lim}\limits_{x \rightarrow -\infty} f(x) = 0\), \(\operatorname*{lim}\limits_{x \rightarrow 0^+} f(x) = \infty\), \(\operatorname*{lim}\limits_{x \rightarrow 0^-} f(x) = -\infty\)

8 Limits at Infinity - Sketching · Level 3

Sketch the graph of an example of a function \(f\) that satisfies all of the given conditions. \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x) = 3\), \(\operatorname*{lim}\limits_{x \rightarrow 2^-} f(x) = \infty\), \(\operatorname*{lim}\limits_{x \rightarrow 2^+} f(x) = -\infty\), \(f\) is odd

9 Limits at Infinity - Sketching · Level 3

Sketch the graph of an example of a function \(f\) that satisfies all of the given conditions. \(f(0) = 3\), \(\operatorname*{lim}\limits_{x \rightarrow 0^-} f(x) = 4\), \(\operatorname*{lim}\limits_{x \rightarrow 0^+} f(x) = 2\), \(\operatorname*{lim}\limits_{x \rightarrow -\infty} f(x) = -\infty\), \(\operatorname*{lim}\limits_{x \rightarrow 4^-} f(x) = -\infty\), \(\operatorname*{lim}\limits_{x \rightarrow 4^+} f(x) = \infty\), \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x) = 3\)

10 Limits at Infinity - Sketching · Level 3

Sketch the graph of an example of a function \(f\) that satisfies all of the given conditions. \(\operatorname*{lim}\limits_{x \rightarrow 3} f(x) = -\infty\), \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x) = 2\), \(f(0) = 0\), \(f\) is even

11 Limits at Infinity - Numerical · Level 2

Guess the value of the limit \( \operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{x^2}{2^x} \) by evaluating the function \(f(x) = \dfrac{x^2}{2^x}\) for \(x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 50\), and \(100\). Then use a graph of \(f\) to support your guess.

12 Limits at Infinity - Numerical · Level 3

(a) Use a graph of \( f(x) = \left(1 - \dfrac{2}{x}\right)^x \) to estimate the value of \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x)\) correct to two decimal places.

Answer for 12(a)

(b) Use a table of values of \(f(x)\) to estimate the limit to four decimal places.

Answer for 12(b)

Enter your answer directly below each part above.

13 Limits at Infinity - Rational Functions · Level 2

Evaluate the limit and justify each step by indicating the appropriate properties of limits. \( \operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{2x^2 - 7}{5x^2 + x - 3} \)

14 Limits at Infinity - Rational Functions · Level 3

Evaluate the limit and justify each step by indicating the appropriate properties of limits. \( \operatorname*{lim}\limits_{x \rightarrow \infty} \sqrt{\dfrac{9x^3 + 8x - 4}{3 - 5x + x^3}} \)

15 Limits at Infinity - Computation · Level 2

Find the limit. \( \operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{3x - 2}{2x + 1} \)

16 Limits at Infinity - Computation · Level 2

Find the limit. \( \operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{1 - x^2}{x^3 - x + 1} \)

17 Limits at Infinity - Computation · Level 2

Find the limit. \( \operatorname*{lim}\limits_{x \rightarrow -\infty} \dfrac{x - 2}{x^2 + 1} \)

18 Limits at Infinity - Computation · Level 2

Find the limit. \( \operatorname*{lim}\limits_{x \rightarrow -\infty} \dfrac{4x^3 + 6x^2 - 2}{2x^3 - 4x + 5} \)

19 Limits at Infinity - Computation · Level 3

Find the limit. \( \operatorname*{lim}\limits_{t \rightarrow \infty} \dfrac{\sqrt{t} + t^2}{2t - t^2} \)

20 Limits at Infinity - Computation · Level 3

Find the limit. \( \operatorname*{lim}\limits_{t \rightarrow \infty} \dfrac{t - t \sqrt{t}}{2t^{\dfrac{3}{2}} + 3t - 5} \)

21 Limits at Infinity - Computation · Level 3

Find the limit. \( \operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{(2x^2 + 1)^2}{(x - 1)^2 (x^2 + x)} \)

22 Limits at Infinity - Computation · Level 3

Find the limit. \( \operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{x^2}{\sqrt{x^4 + 1}} \)

23 Limits at Infinity - Computation · Level 3

Find the limit. \( \operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{\sqrt{1 + 4x^6}}{2 - x^3} \)

24 Limits at Infinity - Computation · Level 3

Find the limit. \( \operatorname*{lim}\limits_{x \rightarrow -\infty} \dfrac{\sqrt{1 + 4x^6}}{2 - x^3} \)

25 Limits at Infinity - Computation · Level 3

Find the limit. \( \operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{\sqrt{x + 3x^2}}{4x - 1} \)

26 Limits at Infinity - Computation · Level 3

Find the limit. \( \operatorname*{lim}\limits_{x \rightarrow -\infty} \dfrac{x + 3x^2}{4x - 1} \)

27 Limits at Infinity - Radical Expressions · Level 4

Find the limit. \( \operatorname*{lim}\limits_{x \rightarrow \infty} (\sqrt{9x^2 + x} - 3x) \)

28 Limits at Infinity - Radical Expressions · Level 4

Find the limit. \( \operatorname*{lim}\limits_{x \rightarrow -\infty} (\sqrt{4x^2 + 3x} + 2x) \)

29 Limits at Infinity - Radical Expressions · Level 4

Find the limit. \( \operatorname*{lim}\limits_{x \rightarrow \infty} (\sqrt{x^2 + a x} - \sqrt{x^2 + b x}) \)

30 Limits at Infinity - Computation · Level 3

Find the limit or show that it does not exist. \( \operatorname*{lim}\limits_{x \rightarrow \infty} (x - \sqrt{x}) \)

31 Limits at Infinity - Computation · Level 3

Find the limit. \( \operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{x^4 - 3x^2 + x}{x^3 - x + 2} \)

32 Limits at Infinity - Computation · Level 3

Find the limit. \( \operatorname*{lim}\limits_{x \rightarrow \infty} (e^{-x} + 2 \cos 3x) \)

33 Limits at Infinity - Computation · Level 2

Find the limit. \( \operatorname*{lim}\limits_{x \rightarrow -\infty} (x^2 + 2x^7) \)

34 Limits at Infinity - Computation · Level 3

Find the limit. \( \operatorname*{lim}\limits_{x \rightarrow -\infty} \dfrac{1 + x^6}{x^4 + 1} \)

35 Limits at Infinity - Transcendental · Level 3

Find the limit. \( \operatorname*{lim}\limits_{x \rightarrow \infty} \arctan(e^x) \)

36 Limits at Infinity - Transcendental · Level 3

Find the limit. \( \operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{e^{3x} - e^{-3x}}{e^{3x} + e^{-3x}} \)

37 Limits at Infinity - Transcendental · Level 3

Find the limit. \( \operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{1 - e^x}{1 + 2 e^x} \)

38 Limits at Infinity - Transcendental · Level 3

Find the limit. \( \operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{\sin^2 x}{x^2 + 1} \)

39 Limits at Infinity - Transcendental · Level 3

Find the limit. \( \operatorname*{lim}\limits_{x \rightarrow \infty} (e^{-2x} \cos x) \)

40 Limits at Infinity - Transcendental · Level 3

Find the limit. \( \operatorname*{lim}\limits_{x \rightarrow 0^+} \arctan(\ln x) \)

41 Limits at Infinity - Logarithmic · Level 3

Find the limit. \( \operatorname*{lim}\limits_{x \rightarrow \infty} [\ln(1 + x^2) - \ln(1 + x)] \)

42 Limits at Infinity - Logarithmic · Level 3

Find the limit. \( \operatorname*{lim}\limits_{x \rightarrow \infty} [\ln(2 + x) - \ln(1 + x)] \)

43 Limits at Infinity - Multi-part · Level 4

(a) For \(f(x) = \dfrac{x}{\ln x}\), find each of the following limits. (i) \(\operatorname*{lim}\limits_{x \rightarrow 0^+} f(x)\) (ii) \(\operatorname*{lim}\limits_{x \rightarrow 1^-} f(x)\) (iii) \(\operatorname*{lim}\limits_{x \rightarrow 1^+} f(x)\)

Answer for 43(a)

(b) Use a table of values to estimate \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x)\).

Answer for 43(b)

(c) Use the information from parts (a) and (b) to make a rough sketch of the graph of \(f\).

Answer for 43(c)

Enter your answer directly below each part above.

44 Limits at Infinity - Multi-part · Level 4

For \(f(x) = \dfrac{2}{x} - \dfrac{1}{\ln x}\), find each of the following limits.

(a) \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x)\)

Answer for 44(a)

(b) \(\operatorname*{lim}\limits_{x \rightarrow 0^+} f(x)\)

Answer for 44(b)

(c) \(\operatorname*{lim}\limits_{x \rightarrow 1^-} f(x)\)

Answer for 44(c)

(d) \(\operatorname*{lim}\limits_{x \rightarrow 1^+} f(x)\)

Answer for 44(d)

(e) Use the information from parts (a)-(d) to make a rough sketch of the graph of \(f\).

Answer for 44(e)

Enter your answer directly below each part above.

45 Limits at Infinity - Estimation · Level 3

(a) Estimate the value of \( \operatorname*{lim}\limits_{x \rightarrow -\infty} (\sqrt{x^2 + x + 1} + x) \) by graphing the function \(f(x) = \sqrt{x^2 + x + 1} + x\).

Answer for 45(a)

(b) Use a table of values of \(f(x)\) to guess the value of the limit.

Answer for 45(b)

(c) Prove that your guess is correct.

Answer for 45(c)

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46 Limits at Infinity - Estimation · Level 3

(a) Use a graph of \( f(x) = \sqrt{3x^2 + 8x + 6} - \sqrt{3x^2 + 3x + 1} \) to estimate the value of \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x)\) to one decimal place.

Answer for 46(a)

(b) Use a table of values of \(f(x)\) to estimate the limit to four decimal places.

Answer for 46(b)

(c) Find the exact value of the limit.

Answer for 46(c)

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47 Limits at Infinity - Asymptotes · Level 2

Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. \( y = \dfrac{5 + 4x}{x + 3} \)

48 Limits at Infinity - Asymptotes · Level 3

Find the horizontal and vertical asymptotes of each curve. \( y = \dfrac{2x^2 + 1}{3x^2 + 2x - 1} \)

49 Limits at Infinity - Asymptotes · Level 3

Find the horizontal and vertical asymptotes of each curve. \( y = \dfrac{2x^2 + x - 1}{x^2 + x - 2} \)

50 Limits at Infinity - Asymptotes · Level 3

Find the horizontal and vertical asymptotes of each curve. \( y = \dfrac{1 + x^4}{x^2 - x^4} \)

51 Limits at Infinity - Asymptotes · Level 3

Find the horizontal and vertical asymptotes of each curve. \( y = \dfrac{x^3 - x}{x^2 - 6x + 5} \)

52 Limits at Infinity - Asymptotes · Level 3

Find the horizontal and vertical asymptotes of each curve. \( y = \dfrac{2 e^x}{e^x - 5} \)

53 Limits at Infinity - Graphical · Level 3

Estimate the horizontal asymptote of the function \( f(x) = \dfrac{3x^3 + 500x^2}{x^3 + 500x^2 + 100x + 2000} \) by graphing \(f\) for \(-10 \leq x \leq 10\). Then calculate the equation of the asymptote by evaluating the limit. How do you explain the discrepancy?

54 Limits at Infinity - Graphical · Level 4

(a) Graph the function \( f(x) = \dfrac{\sqrt{2x^2 + 1}}{3x - 5} \) How many horizontal and vertical asymptotes do you observe? Use the graph to estimate the values of the limits \( \operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{\sqrt{2x^2 + 1}}{3x - 5} \) and \( \operatorname*{lim}\limits_{x \rightarrow -\infty} \dfrac{\sqrt{2x^2 + 1}}{3x - 5} \)

Answer for 54(a)

(b) By calculating values of \(f(x)\), give numerical estimates of the limits in part (a).

Answer for 54(b)

(c) Calculate the exact values of the limits in part (a). Did you get the same value or different values for these two limits? [In view of your answer to part (a), you might have to check your calculation for the second limit.]

Answer for 54(c)

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55 Limits at Infinity - Theory · Level 3

Let \(P\) and \(Q\) be polynomials. Find \( \operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{P(x)}{Q(x)} \) if the degree of \(P\) is (a) less than the degree of \(Q\) and (b) greater than the degree of \(Q\).

56 Limits at Infinity - Power Functions · Level 3

Make a rough sketch of the curve \(y = x^n\) (\(n\) an integer) for the following five cases: (i) \(n = 0\) (ii) \(n > 0\), \(n\) odd (iii) \(n > 0\), \(n\) even (iv) \(n < 0\), \(n\) odd (v) \(n < 0\), \(n\) even Then use these sketches to find the following limits.

(a) \(\operatorname*{lim}\limits_{x \rightarrow 0^+} x^n\)

Answer for 56(a)

(b) \(\operatorname*{lim}\limits_{x \rightarrow 0^-} x^n\)

Answer for 56(b)

(c) \(\operatorname*{lim}\limits_{x \rightarrow \infty} x^n\)

Answer for 56(c)

(d) \(\operatorname*{lim}\limits_{x \rightarrow -\infty} x^n\)

Answer for 56(d)

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57 Limits at Infinity - Function Design · Level 4

Find a formula for a function \(f\) that satisfies the following conditions: \(\operatorname*{lim}\limits_{x \rightarrow \pm \infty} f(x) = 0\), \(\operatorname*{lim}\limits_{x \rightarrow 0} f(x) = -\infty\), \(f(2) = 0\), \(\operatorname*{lim}\limits_{x \rightarrow 3^-} f(x) = \infty\), \(\operatorname*{lim}\limits_{x \rightarrow 3^+} f(x) = -\infty\)

58 Limits at Infinity - Function Design · Level 3

Find a formula for a function that has vertical asymptotes \(x = 1\) and \(x = 3\) and horizontal asymptote \(y = 1\).

59 Limits at Infinity - Function Design · Level 4

A function \(f\) is a ratio of quadratic functions and has a vertical asymptote \(x = 4\) and just one \(x\)-intercept, \(x = 1\). It is known that \(f\) has a removable discontinuity at \(x = -1\) and \(\operatorname*{lim}\limits_{x \rightarrow -1} f(x) = 2\). Evaluate

(a) \(f(0)\)

Answer for 59(a)

(b) \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x)\)

Answer for 59(b)

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60 Limits at Infinity - End Behavior · Level 3

Find the limits as \(x \rightarrow \infty\) and as \(x \rightarrow -\infty\). Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12. \( y = 2x^3 - x^4 \)

61 Limits at Infinity - End Behavior · Level 3

Find the limits as \(x \rightarrow \infty\) and as \(x \rightarrow -\infty\). Use this information, together with intercepts, to give a rough sketch of the graph. \( y = x^4 - x^6 \)

62 Limits at Infinity - End Behavior · Level 3

Find the limits as \(x \rightarrow \infty\) and as \(x \rightarrow -\infty\). Use this information, together with intercepts, to give a rough sketch of the graph. \( y = x^3 (x + 2)^2 (x - 1) \)

63 Limits at Infinity - End Behavior · Level 3

Find the limits as \(x \rightarrow \infty\) and as \(x \rightarrow -\infty\). Use this information, together with intercepts, to give a rough sketch of the graph. \( y = (3 - x)(1 + x)^2 (1 - x)^4 \)

64 Limits at Infinity - End Behavior · Level 3

Find the limits as \(x \rightarrow \infty\) and as \(x \rightarrow -\infty\). Use this information, together with intercepts, to give a rough sketch of the graph. \( y = x^2 (x^2 - 1)^2 (x + 2) \)

65 Limits at Infinity - Squeeze Theorem · Level 3

(a) Use the Squeeze Theorem to evaluate \(\operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{\sin x}{x}\).

Answer for 65(a)

(b) Graph \(f(x) = (\sin x) / x\). How many times does the graph cross the asymptote?

Answer for 65(b)

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66 Limits at Infinity - End Behavior · Level 3

By the end behavior of a function we mean the behavior of its values as \(x \rightarrow \infty\) and as \(x \rightarrow -\infty\).

(a) Describe and compare the end behavior of the functions \(P(x) = 3x^5 - 5x^3 + 2x\) and \(Q(x) = 3x^5\) by graphing both functions in the viewing rectangles \([-2, 2]\) by \([-2, 2]\) and \([-10, 10]\) by \([-10000, 10000]\).

Answer for 66(a)

(b) Two functions are said to have the same end behavior if their ratio approaches 1 as \(x \rightarrow \infty\). Show that \(P\) and \(Q\) have the same end behavior.

Answer for 66(b)

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67 Limits at Infinity - Squeeze Theorem · Level 3

Find \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x)\) if, for all \(x > 1\), \( \dfrac{10 e^x - 21}{2 e^x} < f(x) < \dfrac{5 \sqrt{x}}{\sqrt{x - 1}} \)

68 Limits at Infinity - Applications · Level 3

(a) A tank contains 5000 L of pure water. Brine that contains 30 g of salt per liter of water is pumped into the tank at a rate of 25 L/min. Show that the concentration of salt after \(t\) minutes (in grams per liter) is \( C(t) = \dfrac{30t}{200 + t} \)

Answer for 68(a)

(b) What happens to the concentration as \(t \rightarrow \infty\)?

Answer for 68(b)

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69 Limits at Infinity - Applications · Level 3

In Chapter 9 we will be able to show, under certain assumptions, that the velocity \(v(t)\) of a falling raindrop at time \(t\) is \( v(t) = v^* (1 - e^{-g t / v^*}) \) where \(g\) is the acceleration due to gravity and \(v^*\) is the terminal velocity of the raindrop.

(a) Find \(\operatorname*{lim}\limits_{t \rightarrow \infty} v(t)\).

Answer for 69(a)

(b) Graph \(v(t)\) if \(v^* = 1\) m/s and \(g = 9.8\) m/s\({}^2\). How long does it take for the velocity of the raindrop to reach 99% of its terminal velocity?

Answer for 69(b)

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70 Limits at Infinity - Applications · Level 3

(a) By graphing \(y = e^{-\dfrac{x}{10}}\) and \(y = 0.1\) on a common screen, discover how large you need to make \(x\) so that \(e^{-\dfrac{x}{10}} < 0.1\).

Answer for 70(a)

(b) Can you solve part (a) without using a graphing device?

Answer for 70(b)

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71 Limits at Infinity - Epsilon-N · Level 4

Use a graph to find a number \(N\) such that if \(x > N\) then \(|\dfrac{3x^2 + 1}{2x^2 + x + 1} - 1.5| < 0.05\)

72 Limits at Infinity - Epsilon-N · Level 4

For the limit \( \operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{1 - 3x}{\sqrt{x^2 + 1}} = -3 \) illustrate Definition 7 by finding values of \(N\) that correspond to \(\epsilon = 0.1\) and \(\epsilon = 0.05\).

73 Limits at Infinity - Epsilon-N · Level 4

For the limit \( \operatorname*{lim}\limits_{x \rightarrow -\infty} \dfrac{1 - 3x}{\sqrt{x^2 + 1}} = 3 \) illustrate Definition 8 by finding values of \(N\) that correspond to \(\epsilon = 0.1\) and \(\epsilon = 0.05\).

74 Limits at Infinity - Epsilon-N · Level 4

For the limit \( \operatorname*{lim}\limits_{x \rightarrow \infty} \sqrt{x} \ln x = \infty \) illustrate Definition 9 by finding a value of \(N\) that corresponds to \(M = 100\).

75 Limits at Infinity - Proofs · Level 4

(a) How large do we have to take \(x\) so that \(1 / x^2 < 0.0001\)?

Answer for 75(a)

(b) Taking \(r = 2\) in Theorem 5, we have the statement \( \operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{1}{x^2} = 0 \) Prove this directly using Definition 7.

Answer for 75(b)

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76 Limits at Infinity - Proofs · Level 4

(a) How large do we have to take \(x\) so that \(\dfrac{1}{\sqrt{x}} < 0.0001\)?

Answer for 76(a)

(b) Taking \(r = \dfrac{1}{2}\) in Theorem 5, we have the statement \( \operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{1}{\sqrt{x}} = 0 \) Prove this directly using Definition 7.

Answer for 76(b)

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77 Limits at Infinity - Proofs · Level 5

Use Definition 8 to prove that \(\operatorname*{lim}\limits_{x \rightarrow -\infty} \dfrac{1}{x} = 0\).

78 Limits at Infinity - Proofs · Level 5

Prove, using Definition 9, that \(\operatorname*{lim}\limits_{x \rightarrow \infty} x^3 = \infty\).

79 Limits at Infinity - Proofs · Level 5

Use Definition 9 to prove that \(\operatorname*{lim}\limits_{x \rightarrow \infty} e^x = \infty\).

80 Limits at Infinity - Proofs · Level 5

Formulate a precise definition of \( \operatorname*{lim}\limits_{x \rightarrow -\infty} f(x) = -\infty \) Then use your definition to prove that \( \operatorname*{lim}\limits_{x \rightarrow -\infty} (1 + x^3) = -\infty \)

81 Limits at Infinity - Proofs · Level 5

(a) Prove that \( \operatorname*{lim}\limits_{x \rightarrow \infty} f(x) = \operatorname*{lim}\limits_{t \rightarrow 0^+} f\left(\dfrac{1}{t}\right) \) and \( \operatorname*{lim}\limits_{x \rightarrow -\infty} f(x) = \operatorname*{lim}\limits_{t \rightarrow 0^-} f\left(\dfrac{1}{t}\right) \) if these limits exist.

Answer for 81(a)

(b) Use part (a) and Exercise 65 to find \( \operatorname*{lim}\limits_{x \rightarrow 0^+} x \sin \dfrac{1}{x} \)

Answer for 81(b)

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