Section 2.6 Continuity 2.6 Continuity The graphs of many functions encountered in this text contain no holes, jumps, or breaks. For example, if L = f HtL represents the length of a fish t years after it is hatched, then the length of the fish changes gradually as t increases. Consequently, the graph of L = f HtL contains no breaks (Figure 2.46a). Some functions, however, do contain abrupt changes in their values. Consider a parking meter that accepts only quarters and each quarter buys 15 minutes of parking. Letting cHtL be the cost (in dollars) of parking for t minutes, the graph of c has breaks at integer multiples of 15 minutes (Figure 2.46b). Figure 2.46 Informally, we say that a function f is continuous at a if the graph of f does not have a hole or break at a (that is, if the graph near a can be drawn without lifting the pencil). If a function is not continuous at a, then a is a point of discontinuity. QUICK CHECK 1 For what values of t in H0, 60L does the graph of y = cHtL in Figure 2.46b have discontinuities? Continuity at a Point Continuity on an Interval Functions Involving Roots Continuity of Transcendental Functions Intermediate Value Theorem Quick Quiz SECTION 2.6 EXERCISES Review Questions 1. Which of the following functions are continuous for all values in their domain? Justify your answers. a. aHtL = altitude of a skydiver t seconds after jumping from a plane b. nHtL = number of quarters needed to park in a metered parking space for t minutes c. THtL = temperature t minutes after midnight in Chicago on January 1 d. pHtL = number of points scored by a basketball player after t minutes of a basketball game 2. Give the three conditions that must be satisfied by a function to be continuous at a point. 3. What does it mean for a function to be continuous on an interval? Copyright © 2014 Pearson Education, Inc. 1 2 Chapter 2 • Limits 4. We informally described a function f to be continuous at a if its graph contains no holes or breaks at a. Explain why this is not an adequate definition of continuity. 5. Complete the following sentences. a. A function is continuous from the left at a if _______. b. A function is continuous from the right at a if _______. 6. Describe the points (if any) at which a rational function fails to be continuous. 7. What is the domain of f HxL = 8. ex and where is f continuous? x Explain the Intermediate Value Theorem using words and pictures. Basic Skills 9–12. Discontinuities from a graph Determine the points at which the following functions f have discontinuities. At each point of discontinuity, state the conditions in the continuity checklist that are violated. 9. 10. 11. Copyright © 2014 Pearson Education, Inc. Section 2.6 Continuity 12. 13–20. Continuity at a point Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. 13. f HxL = 14. f HxL = 2 x2 + 3 x + 1 x2 + 5 x 2 x2 + 3 x + 1 x2 + 5 x 15. f HxL = 16. gHxL = ; a = -5 x-2 ; a=1 1 x-3 ; a=3 x2 - 1 17. f HxL = ; a=5 x-1 3 if x ∫ 1 if x = 1 x2 - 4 x + 3 18. f HxL = x-3 5x-2 x2 if x ∫ 3 ; a=3 if x = 3 2 19. f HxL = ; a=1 ; a=4 - 9 x + 20 Copyright © 2014 Pearson Education, Inc. 3 4 Chapter 2 • Limits x2 + x 20. f HxL = x+1 2 if x ∫ -1 ; a = -1 if x = -1 21–26. Continuity on intervals Use Theorem 2.10 to determine the intervals on which the following functions are continuous. 21. pHxL = 4 x5 - 3 x2 + 1 22. gHxL = 23. f HxL = 3 x2 - 6 x + 7 x2 + x + 1 x5 + 6 x + 17 x2 - 9 x2 - 4 x + 3 24. sHxL = x2 - 1 25. f HxL = 26. f HtL = 1 x2 -4 t+2 t2 - 4 27–30. Limits of compositions Evaluate the following limits and justify your answers. 27. lim Ix8 - 3 x6 - 1M40 xØ0 28. lim xØ2 29. lim xØ1 4 3 2 x5 - 4 x2 - 50 x+5 4 x+2 2x+1 30. lim xØ¶ 3 x 31–34. Limits of composite functions Evaluate each limit and justify your answer. x3 - 2 x2 - 8 x 31. lim x-4 xØ4 32. lim t-4 tØ4 33. lim ln xØ0 t -2 2 sin x x Copyright © 2014 Pearson Education, Inc. Section 2.6 Continuity xØ0 1ê3 x 34. lim 16 x + 1 - 1 35–38. Intervals of continuity Determine the intervals of continuity for the following functions. 35. The graph of Exercise 9 36. The graph of Exercise 10 37. The graph of Exercise 11 38. The graph of Exercise 12 39. Intervals of continuity Let 2x f HxL = if x < 1 2 x + 3 x if x ¥ 1. a. Use the continuity checklist to show that f is not continuous at 1. b. Is f continuous from the left or right at 1? c. State the interval(s) of continuity. 40. Intervals of continuity Let f HxL = x3 + 4 x + 1 if x 0 2 x3 if x > 0. a. Use the continuity checklist to show that f is not continuous at 0. b. Is f continuous from the left or right at 0? c. State the interval(s) of continuity. 41–46. Functions with roots Determine the interval(s) on which the following functions are continuous. Be sure to consider right- and left-continuity at the endpoints. 41. f HxL = 2 x2 - 16 42. gHxL = x4 - 1 43. f HxL = 3 x2 - 2 x - 3 44. f HtL = It2 - 1M3ê2 45. f HxL = H2 x - 3L2ê3 46. f HzL = Hz - 1L3ê4 47–50. Limits with roots Evaluate each limit and justify your answers. 47. lim xØ2 4 x + 10 2x-2 48. lim x2 - 4 + 3 x2 - 9 xØ-1 Copyright © 2014 Pearson Education, Inc. 5 6 Chapter 2 • Limits 49. lim x2 + 7 xØ3 t2 + 5 50. lim tØ2 t2 + 5 1+ 51–56. Continuity and limits with transcendental functions Determine the interval(s) on which the following functions are continuous; then evaluate the given limits. 51. f HxL = csc x; 52. f HxL = e 53. f HxL = 54. f HxL = 55. f HxL = 56. f HxL = T x lim f HxL; xØpê4 lim f HxL; ; xØ4 1 + sin x ; cos x ln x sin-1 x ex 1 - ex ; ex - 1 lim f HxL xØ0+ lim f HxL; xØpê2- lim f HxL xØ4 pê3 lim f HxL xØ1- lim f HxL; lim f HxL ; e2 x - 1 lim f HxL xØ2 p- xØ0- ; xØ0+ lim f HxL xØ0 57. Intermediate Value Theorem and interest rates Suppose $5000 is invested in a savings account for 10 years (120 months), with an annual interest rate of r, compounded monthly. The amount of money in the account after r 120 10 years is AHrL = 5000 1 + . 12 a. Use the Intermediate Value Theorem to show there is a value of r in H0, 0.08L—an interest rate between 0% and 8%—that allows you to reach your savings goal of $7000 in 10 years. b. Use a graph to illustrate your explanation in part (a); then, approximate the interest rate required to reach your goal. T 58. Intermediate Value Theorem and mortgage payments You are shopping for a $150,000, 30-year (360-month) loan to buy a house. The monthly payment is 150, 000 Hrê 12L mHrL = , 1 - H1 + rê 12L-360 where r is the annual interest rate. Suppose banks are currently offering interest rates between 6% and 8%. a. Use the Intermediate Value Theorem to show there is a value of r in H0.06, 0.08L—an interest rate between 6% and 8%—that allows you to make monthly payments of $1000 per month. b. Use a graph to illustrate your explanation to part (a). Then determine the interest rate you need for monthly payments of $1000. T 59–64. Applying the Intermediate Value Theorem a. Use the Intermediate Value Theorem to show that the following equations have a solution on the given interval. b. Use a graphing utility to find all the solutions to the equation on the given interval. Copyright © 2014 Pearson Education, Inc. Section 2.6 Continuity c. Illustrate your answers with an appropriate graph. 59. 2 x3 + x - 2 = 0; 60. H-1, 1L x4 + 25 x3 + 10 = 5; 61. x3 - 5 x2 + 2 x = -1; 62. -x5 - 4 x2 + 2 63. x + ex = 0; H0, 1L H-1, 5L x + 5 = 0; H0, 3L H-1, 0L 64. x ln x - 1 = 0; H1, eL Further Explorations 65. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If a function is left-continuous and right-continuous at a, then it is continuous at a. b. If a function is continuous at a, then it is left-continuous and right-continuous at a. c. If a < b and f HaL L f HbL, then there is some value of c between a and b for which f HcL = L. d. Suppose f is continuous on @a, bD. Then there is a point c in Ha, bL such that f HcL = f HaL + f HbL . 2 66. Continuity of the absolute value function Prove that the absolute value function †x§ is continuous for all values of x. (Hint: Using the definition of the absolute value function, compute lim- †x§ and lim †x§.) xØ0 xØ0+ 67–70. Continuity of functions with absolute values Use the continuity of the absolute value function (Exercise 66) to determine the interval(s) on which the following functions are continuous. 67. f HxL = °x2 + 3 x - 18• 68. gHxL = 69. hHxL = x+4 x2 - 4 1 x -4 70. hHxL = °x2 + 2 x + 5• + x 71–80. Miscellaneous limits Evaluate the following limits or state that they do not exist. cos2 x + 3 cos x + 2 71. lim xØp 72. lim xØ3 pê2 73. lim xØpê2 cos x + 1 sin2 x + 6 sin x + 5 sin2 x - 1 sin x - 1 sin x - 1 Copyright © 2014 Pearson Education, Inc. 7 8 Chapter 2 • Limits 1 2+sin q 74. lim qØ0 - 1 2 sin q cos x - 1 75. lim sin2 x xØ0 1 - cos2 x 76. lim xØ0+ sin x tan-1 x 77. lim xØ¶ 78. lim x cos t tØ¶ 79. limxØ1 e3 t x ln x x 80. lim xØ0+ ln x 81. Pitfalls using technology The graph of the sawtooth function y = x - dxt, where dxt is the greatest integer function or floor function (Exercise 37, Section 2.2) was obtained using a graphing utility (see figure). Identify any inaccuracies appearing in the graph and then plot an accurate graph by hand. T 82. Pitfalls using technology Graph the function f HxL = sin x x using a graphing window of @-p, pD µ @0, 2D. a. Sketch a copy of the graph obtained with your graphing device and describe any inaccuracies appearing in the graph. b. Sketch an accurate graph of the function. Is f continuous at 0? c. Conjecture the value of lim xØ0 sin x . x 83. Sketching functions a. Sketch the graph of a function that is not continuous at 1, but is defined at 1. b. Sketch the graph of a function that is not continuous at 1, but has a limit at 1. Copyright © 2014 Pearson Education, Inc. Section 2.6 Continuity 84. An unknown constant Determine the value of the constant a for which the function x2 + 3 x + 2 f HxL = x+1 a if x ∫ -1 if x = -1 is continuous at -1. 85. An unknown constant Let gHxL = x2 + x if x < 1 a if x = 1 3 x + 5 if x > 1 a. Determine the value of a for which g is continuous from the left at 1. b. Determine the value of a for which g is continuous from the right at 1. c. Is there a value of a for which g is continuous at 1? Explain. T 86. Asymptotes of a function containing exponentials Let f HxL = 2 ex + 5 e3 x . Analyze e2 x - e3 x lim- f HxL, lim f HxL, lim f HxL, and lim f HxL. Then give the horizontal and vertical asymptotes of f . Plot f to xØ0 xØ0+ xØ-¶ xØ¶ verify your results. T 87. Asymptotes of a function containing exponentials Let f HxL = 2 ex + 10 e-x . Analyze e x + e- x lim f HxL, lim f HxL, and lim f HxL. Then give the horizontal and vertical asympotes of f . Plot f to verify your xØ0 xØ-¶ xØ¶ results. T 88–89. Applying the Intermediate Value Theorem Use the Intermediate Value Theorem to verify that the following equations have three solutions on the given interval. Use a graphing utility to find the approximate roots. 88. x3 + 10 x2 - 100 x + 50 = 0; H-20, 10L 89. 70 x3 - 87 x2 + 32 x - 3 = 0; H0, 1L Applications 90. Parking costs Determine the intervals of continuity for the parking cost function c introduced at the outset of this section (see the figure). Consider 0 t 60. Copyright © 2014 Pearson Education, Inc. 9 10 Chapter 2 • Limits T 91. Investment problem Assume you invest $250 at the end of each year for 10 years at an annual interest rate of r. The amount of money in your account after 10 years is AHrL = 250 IH1 + rL10 - 1M . Assume your goal is to have r $3500 in your account after 10 years. a. Use the Intermediate Value Theorem to show that there is an interest rate r in the interval H0.01, 0.10L— between 1% and 10%—that allows you to reach your financial goal. b. Use a calculator to estimate the interest rate required to reach your financial goal. 92. Applying the Intermediate Value Theorem Suppose you park your car at a trailhead in a national park and begin a 2-hr hike to a lake at 7 A.M. on a Friday morning. On Sunday morning, you leave the lake at 7 A.M. and start the 2-hr hike back to your car. Assume the lake is 3 mi from your car. Let f HtL be your distance from the car t hours after 7 A.M. on Friday morning and let gHtL be your distance from the car t hours after 7 A.M. on Sunday morning. a. Evaluate f H0L, f H2L, gH0L, and gH2L. b. Let hHtL = f HtL - gHtL. Find hH0L and hH2L. c. Use the Intermediate Value Theorem to show that there is some point along the trail that you will pass at exactly the same time of morning on both days. 93. The monk and the mountain A monk set out from a monastery in the valley at dawn. He walked all day up a winding path, stopping for lunch and taking a nap along the way. At dusk, he arrived at a temple on the mountaintop. The next day, the monk made the return walk to the valley, leaving the temple at dawn, walking the same path for the entire day, and arriving at the monastery in the evening. Must there be one point along the path that the monk occupied at the same time of day on both the ascent and descent? (Hint: The question can be answered without the Intermediate Value Theorem.) (Source: Arthur Koestler, The Act of Creation) Additional Exercises 94. Does continuity of † f § imply continuity of f ? Let gHxL = 1 if x ¥ 0 -1 if x < 0. a. Write a formula for †gHxL§. b. Is g continuous at x = 0? Explain. c. Is †g§ continuous at x = 0? Explain. d. For any function f , if † f § is continuous at a, does it necessarily follow that f is continuous at a? Explain. 95–96. Classifying discontinuities The discontinuities in graphs (a) and (b) are removable discontinuities because they disappear if we define or redefine f at a so that f HaL = lim f HxL. The function in graph (c) has a jump xØa discontinuity because left and right limits exist at a but are unequal. The discontinuity in graph (d) is an infinite discontinuity because the function has a vertical asymptote at a. Copyright © 2014 Pearson Education, Inc. Section 2.6 Continuity 95. Is the discontinuity at a in graph (c) removable? Explain. 96. Is the discontinuity at a in graph (d) removable? Explain. 97–98. Removable discontinuities Show that the following functions have a removable discontinuity at the given point. See Exercises 95–96. 97. f HxL = x2 - 7 x + 10 x-2 x2 - 1 98. gHxL = 1-x 3 x=2 ; if x ∫ 1 ; x=1 if x = 1 99. Do removable discontinuities exist? Refer to Exercises 95–96. a. Does the function f HxL = x sin b. Does the function gHxL = sin 1 1 have a removable discontinuity at x = 0? x have a removable discontinuity at x = 0? x T 100–101. Classifying discontinuities Classify the discontinuities in the following functions at the given points. See Exercises 95-96. 100. f HxL = 101.hHxL = †x - 2§ x-2 ; x=2 x3 - 4 x2 + 4 x x Hx - 1L ; x = 0 and x = 1 Copyright © 2014 Pearson Education, Inc. 11 12 Chapter 2 • Limits 102.Continuity of composite functions Prove Theorem 2.11: If g is continuous at a and f is continuous at gHaL, then the composition f ë g is continuous at a. (Hint: Write the definition of continuity for f and g separately; then, combine them to form the definition of continuity for f ë g.) 103.Continuity of compositions a. Find functions f and g such that each function is continuous at 0, but f ë g is not continuous at 0. b. Explain why examples satisfying part (a) do not contradict Theorem 2.11. 104.Violation of the Intermediate Value Theorem? Let f HxL = †x§ . Then f H-2L = -1 and f H2L = 1. Therefore, x f H-2L < 0 < f H2L, but there is no value of c between -2 and 2 for which f HcL = 0. Does this fact violate the Intermediate Value Theorem? Explain. 105.Continuity of sin x and cos x a. Use the identity sin Ha + hL = sin a cos h + cos a sin h with the fact that lim sin x = 0 to prove that xØ0 lim sin x = sin a, thereby establishing that sin x is continuous for all x. (Hint: Let h = x - a so that x = a + h xØa and note that h Ø 0 as x Ø a.) b. Use the identity cos Ha + hL = cos a cos h - sin a sin h with the fact that lim cos x = 1 to prove that xØ0 lim cos x = cos a. xØa Copyright © 2014 Pearson Education, Inc.

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