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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
What does it mean for a function to be continuous on an interval?We informally describe 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.Determine the points on the interval (0, 5) at which the following functions f have discontinuities. At each point of discontinuity, state the conditions in the continuity checklist that are violated.Determine the points on the interval (0, 5) at which the following functions f have discontinuities. At each point of discontinuity, state the conditions in the continuity checklist that are violated. 6.Determine the points on the interval (0, 5) at which the following functions f have discontinuities. At each point of discontinuity, state the conditions in the continuity checklist that are violated. 7.Determine the points on the interval (0, 5) at which the following functions f have discontinuities. At each point of discontinuity, state the conditions in the continuity checklist that are violated. 8.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 _____.Evaluate f(3) if limx3f(x)=5,limx3+f(x)=6, and f is right-continuous at x = 3.Determine the intervals of continuity for the following functions. At which endpoints of these intervals of continuity is f continuous from the left or continuous from the right? 11.The graph of Exercise 5 5.Determine the intervals of continuity for the following functions. At which endpoints of these intervals of continuity is f continuous from the left or continuous from the right? 12.The graph of Exercise 6 6.Determine the intervals of continuity for the following functions. At which endpoints of these intervals of continuity is f continuous from the left or continuous from the right? 13.The graph of Exercise 7 7.Determine the intervals of continuity for the following functions. At which endpoints of these intervals of continuity is f continuous from the left or continuous from the right? 14.The graph of Exercise 8 8.What is the domain of f(x) = ex/x and where is f continuous?Parking costs Determine the intervals of continuity for the parking cost function c introduced at the outset of this section (see figure). Consider 0t60.Continuity at a point Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. 14. f(x)=2x2+3x+1x2+5x; a = 5Continuity at a point Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. 13. f(x)=2x2+3x+1x2+5x; a = 5Continuity at a point Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. 15. f(x)=x2; a = 1Continuity at a point Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. 16. g(x)=1x3; a = 3Continuity at a point Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. 17. f(x)={x21x1ifx13ifx=1; a = 1Continuity at a point Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. 18. f(x)={x24x+3x3ifx32ifx=3; a = 3Continuity at a point Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. 19. f(x)=5x2x29x+20; a = 4Continuity at a point Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. 20. f(x)={x2+xx+1ifx12ifx=1; a = 1Continuity on intervals Use Theorem 2.10 to determine the intervals on which the following functions are continuous. 21. p(x) = 4x5 3x2 + 1Continuity on intervals Use Theorem 2.10 to determine the intervals on which the following functions are continuous. 22. g(x)=3x26x+7x2+x+1Continuity on intervals Use Theorem 2.10 to determine the intervals on which the following functions are continuous. 23. f(x)=x5+6x+17x29Continuity on intervals Use Theorem 2.10 to determine the intervals on which the following functions are continuous. 24. s(x)=x24x+3x21Continuity on intervals Use Theorem 2.10 to determine the intervals on which the following functions are continuous. 25. f(x)=1x24Continuity on intervals Use Theorem 2.10 to determine the intervals on which the following functions are continuous. 26.f(t)=t+2t24Limits of compositions Evaluate each limit and justify your answer. 27. limx0(x83x61)40Limits of compositions Evaluate each limit and justify your answer. 28. limx2(32x54x250)4Limits of composite functions Evaluate each limit and justify your answer. 31. limx4x32x28xx4Limits Evaluate each limit and justify your answer. 34.limt4t4t2Limits of compositions Evaluate each limit and justify your answer. 29. limx1(x+5x+2)4Limits of compositions Evaluate each limit and justify your answer. 30. limx(2x+1x)3Limits Evaluate each limit and justify your answer. 37.limx5ln6(x2163)5x25Limits of composite functions Evaluate each limit and justify your answer. 34. limx0(x16x+11)1/3Intervals of continuity Let f(x)={2xifx1x2+3xifx1. 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.Intervals of continuity Let f(x)={x3+4x+1ifx02x3ifx0. 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.Functions with roots Determine the interval(s) on which the following functions are continuous At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right? 41.f(x)=5xFunctions with roots Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right? 42.f(x)=25x2Functions 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(x)=2x216Functions with roots Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right? 44.f(x)=x23x+12Functions with roots Determine the interval(s) on which the following functions are continuous. Be sure to consider right- and left-continuity at the endpoints. 43. f(x)=x22x33Functions with roots Determine the interval(s) on which the following functions are continuous. Be sure to consider right- and left-continuity at the endpoints. 44. f(t)=(t21)3/2Functions with roots Determine the interval(s) on which the following functions are continuous. Be sure to consider right- and left-continuity at the endpoints. 45. f(x)=(2x3)2/3Functions with roots Determine the interval(s) on which the following functions are continuous. Be sure to consider right- and left-continuity at the endpoints. 46. f(z)=(z1)3/4Limits with roots Evaluate each limit and justify your answer. 47. limx24x+102x2Limits with roots Evaluate each limit and justify your answer. 48. limx1(x24+x29)3Miscellaneous limits Evaluate the following limits or state that they do not exist. 71. limxcos2x+3cosx+2cosx+1Miscellaneous limits Evaluate the following limits or state that they do not exist. 72. limx3/2sin2x+6sinx+5sin2x1Limits with roots Evaluate each limit and justify your answer. 49. limx3x2+7Limits with roots Evaluate each limit and justify your answer. 50. limt2t2+51+t2+5Miscellaneous limits Evaluate the following limits or state that they do not exist. 73. limx/2sinx1sinx1Miscellaneous limits Evaluate the following limits or state that they do not exist. 74. lim012+sin12sinMiscellaneous limits Evaluate the following limits or state that they do not exist. 75. limx0cosx1sin2xMiscellaneous limits Evaluate the following limits or state that they do not exist. 76. limx0+1cos2xsinxEvaluate each limit. 59.limx0e4x1ex1Evaluate each limit. 60.limxe2ln2x5lnx+6lnx2Continuity and limits with transcendental functions Determine the interval(s) on which the following functions are continuous; then analyze the given limits. 51. f(x)=cscx;limx/4f(x);limx2f(x)Continuity and limits with transcendental functions Determine the interval(s) on which the following functions are continuous; then analyze the given limits. 52. f(x)=ex;limx4f(x);limx0+f(x)Continuity and limits with transcendental functions Determine the interval(s) on which the following functions are continuous; then analyze the given limits. 53. f(x)=1+sinxcosx;limx/2f(x);limx4/3f(x)Continuity and limits with transcendental functions Determine the interval(s) on which the following functions are continuous; then analyze the given limits. 54. f(x)=lnxsin1x;limx1f(x)Continuity and limits with transcendental functions Determine the interval(s) on which the following functions are continuous; then analyze the given limits. 55. f(x)ex1ex;limx0f(x);limx0+f(x)Continuity and limits with transcendental functions Determine the interval(s) on which the following functions are continuous; then analyze the given limits. 56. f(x)=e2x1ex1;limx0f(x)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. c. Illustrate your answers with an appropriate graph. 59. 2x3+x2=0;(1,1)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. c. Illustrate your answers with an appropriate graph. 60. x4+25x3+10=5;(0,1)Applying the Intermediate Value Theorem a. Use the Intermediate Value Theorem to show that the following equations have a solution on the given interval. a. Use a graphing utility to find all the solutions to the equation on the given interval. b. Illustrate your answers with an appropriate graph. 61. x35x2+2x=1;(1,5)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. c. Illustrate your answers with an appropriate graph. 62. x54x2+2x+5=0;(0,3)Applying the Intermediate Value Theorem a. Use the Intermediate Value Theorem to show that the following equations have a solution on the given interval. a. Use a graphing utility to find all the solutions to the equation on the given interval. b. Illustrate your answers with an appropriate graph. 63. x+ex=0;(1,0)Applying the Intermediate Value Theorem a. Use the Intermediate Value Theorem to show that the following equations have a solution on the given interval. a. Use a graphing utility to find all the solutions to the equation on the given interval. b. Illustrate your answers with an appropriate graph. 64. xlnx1=0;(1,e)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(a) L f(b), then there is some value of c in (a, b) for which f(c) = L. d. Suppose f is continuous on a, b. Then there is a point c in (a, b) such that f(c) = (f(a) + f(b))/2.Mortgage payments You are shopping for a 250,000. 30-year (360-month) loan to buy a house. The monthly payment is given by m(r)=250,000(r/12)1(1+r/12)360, where r is the annual interest rate. Suppose banks are currently offering interest rates between 4% and 5%. a. Show there is a value of r in (0 04, 0.05)an interest rate between 4% and 5%that allows you to make monthly payments of 1300 per month. b. Use a graph to illustrate your explanation to part (a). Then determine the interest rate you need for monthly payments of 1300Intermediate 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 10 years is A(r) = 5000(1 + r/12)120. a. Use the Intermediate Value Theorem to show there is a value of r in (0,0.08)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.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 A=250((1+r)101r. Assume your goal is to have 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 (0.01, 0.10)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.Find an interval containing a solution to the equation 2x=cosx. Use a graphing utility to approximate the solution.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 limx0x and limx0+x.)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(x)=x2+3x18Continuity 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. 68. g(x)=|x+4x24|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. 69. h(x)=|1x4|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. 70. h(x)=|x2+2x+5|+xPitfalls using technology The graph of the sawtooth function y = x x, where x 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.Pitfalls using technology Graph the function f(x)=sinxx using a graphing window of [, ] [0, 2]. 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. What is the value of limx0sinxx?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.An unknown constant Determine the value of the constant a for which the function is continuous at 1. f(x)={x2+3x+2x+1ifx1aifx=1An unknown constant Let g(x)={x2+xifx1aifx=13x+5ifx1. a. Determine the value of a for which g is continuous from the left at 1. b. Determine the value of u 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.Asymptotes of a function containing exponentials Let f(x)=2ex+5e3xe2xe3x. Analyze limx0f(x),limx0+f(x),limxf(x), and limxf(x). Then give the horizontal and vertical asymptotes of f. Plot f to verify your results.Asymptotes of a function containing exponentials Let f(x)=2ex+10exex+ex. Analyze limx0f(x),limxf(x), and limxf(x).Then give the horizontal and vertical asymptotes of f. Plot f to verify your results.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+10x2100x+50=0;(20,10)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. 89. 70x387x2+32x3=0;(0,1)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(t) be your distance from the car t hours after 7 A.M. on Friday morning and let g(t) be your distance from the car t hours after 7 A.M. on Sunday morning. a. Evaluate f(0), f(2), g(0), and g(2). b. Let h(t) = f(t) g(t). Find h(0) and h(2). 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.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.)Does continuity of |f| imply continuity of f? Let g(x)={1ifx01ifx0. a. Write a formula for |g(x)| . 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.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(a)=limxaf(x). The function in graph (c) has a jump 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. 95. Is the discontinuity at a in graph (c) removable? Explain.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(a)=limxaf(x). The function in graph (c) has a jump 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. 96. Is the discontinuity at a in graph (d) removable? Explain.Removable discontinuities Show that the following functions have a removable discontinuity at the given point. See Exercises 9596. 97. f(x)=x27x+10x2;x=2Removable discontinuities Show that the following functions have a removable discontinuity at the given point. See Exercises 9596. 98. g(x)={x211xifx13ifx=1;x=1Classifying discontinuities Classify the discontinuities in the following functions at the given points. See Exercises 9596. 101. h(x)=x34x2+4xx(x1);x=0; x = 0 and x = 1Classifying discontinuities Classify the discontinuities in the following functions at the given points. See Exercises 9596. 100. f(x)=x2x2;x=2Do removable discontinuities exist? See Exercises 9596. a. Does the function f(x) = x sin (1/x) have a removable discontinuity at x = 0? b. Does the function g(x) = sin (1/x) have a removable discontinuity at x = 0?Continuity of composite functions Prove Theorem 2.11: If g is continuous at a and f is continuous at g(a), then the composition f 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 g.)Continuity of compositions a. Find functions f and g such that each function is continuous at 0 but f g is not continuous at 0. b. Explain why examples satisfying part (a) do not contradict Theorem 2.11.Violation of the Intermediate Value Theorem? Let f(x)=xx. Then f(2) = 1 and f(2) = 1. Therefore, f(2) 0 f(2), but there is no value of c between 2 and 2 for which f(c) = 0. Does this fact violate the Intermediate Value Theorem? Explain.Continuity of sin x and cos x a. Use the identity sin (a + h) = sin a cos h + cos a sin h with the fact that limx0sinx=0 to prove that limxasinx=sina, thereby establishing that sin x is continuous for all x. (Hint: Let h = x a so that x = a + h and note that h 0 as x a.) b. Use the identity cos (a + h) = cos a cos h sin a sin h with the fact that limx0cosx=1 to prove that limxacosx=cosa.In Example 1, find a positive number δ satisfying the statement
Example 1 Determining Values of δ from a Graph
Figure 2.59 shows the graph of a linear function f with . For each value of ε > 0, determine a value of δ > 0 satisfying the statement
Figure 2.59
|f(x) − 5| < ε whenever 0 < |x − 3| < δ.
2QCIn Example 7, if N is increased by a factor of 100, how must δ change?
Example 7 An Infinite Limit Proof
Let . Prove that .
Solution
Step 1:
Find δ > 0. Assuming N > 0, we use the inequality to find δ, where δ depends only on N. Taking reciprocals of this inequality, it follows that
Take the square root of both sides.
The inequality has the form |x − 2| < δ if we let . We now write a proof based on this relationship between δ and N.
Step 2:
Write a proof. Suppose N > 0 is given. Let and assume . Squaring both sides of the inequality and taking reciprocals, we have
Square both sides.
Take reciprocals of both sides.
We see that for any positive N, if , then . It follows that . Note that because , δ decreases as N increases.
Suppose x lies in the interval (1, 3) with x 2. Find the smallest positive value of such that the inequality 0 |x 2| is true.Suppose f(x) lies in the interval (2, 6). What is the smallest value of such that |f(x) 4| ?Which one of the following intervals is not symmetric about x = 5? a. (1, 9) b. (4, 6) c. (3, 8) d. (4.5, 5.5)4EState the precise definition of limxaf(x)=L.Interpret |f(x) L| in words.Suppose |f(x) 5| 0.1 whenever 0 x 5. Find all values of 0 such that |f(x) 5| 0.1 whenever 0 |x 2| .Give the definition of limxaf(x)= and interpret it using pictures.Determining values of from a graph The function f in the figure satisfies limx2f(x)=5. Determine the largest value of 0 satisfying each statement a. If 0 |x 2| , then |f(x) 5| 2 . b. If 0 |x 2| , then |f(x) 5| 1.Determining values of from a graph The function f in the figure satisfies limx2f(x)=4. Determine the largest value of 0 satisfying each statement a. If 0 |x 2| , then |f(x) 4| 1. b. If 0 |x 2| , then |f(x) 4| 1/2.Determining values of from a graph The function f in the figure satisfies limx3f(x)=6. Determine the largest value of 0 satisfying each statement. a. If 0 |x 3| , then |f(x) 6| 3. b. If 0 |x 3| , then |f(x) 6| 1.Determining values of from a graph The function f in the figure satisfies limx4f(x)=5. Determine the largest value of 0 satisfying each statement. a. If 0 |x 4| , then |f(x) 5| 1. b. If 0 |x 4| , then |f(x) 5| 0.5.Finding for a given using a graph Let f(x) = x3 + 3 and note that limx0f(x)=3. For each value of , use a graphing utility to find all values of 0 such that |f(x) 3| whenever 0 |x 0| . Sketch graphs illustrating your work. a. = 1 b. = 0.5Finding for a given using a graph Let g(x) = 2x3 12x2 + 26x + 4 and note that limx2g(x)=24. For each value of , use a graphing utility to find all values of 0 such that |g(x) 24| whenever 0 |x 2| . Sketch graphs illustrating your work. a. = 1 b. = 0.5Finding a symmetric interval The function f in the figure satisfies limx2f(x)=3. For each value of , find all values of 0 such that f(x)3whenever0x2. (2) a. = 1 b. =12 c. For any 0, make a conjecture about the corresponding values of satisfying (2).Finding a symmetric interval The function f in the figure satisfies limx4f(x)=5. For each value of , find all values of 0 such that f(x)5whenever0x4. (3) a. = 2 b. = 1 c. For any 0, make a conjecture about the corresponding values of satisfying (3).Finding a symmetric interval Let f(x)=2x22x1 and note that limx1f(x)=4. For each value of , use a graphing utility to find all values of 0 such that |f(x) 4| whenever 0 |x 1| . a. = 2 b. = 1 c. For any 0, make a conjecture about the value of that satisfies the preceding inequality.Finding a symmetric interval Let f(x)={13x+1ifx312x+12ifx3 and note that limx3f(x)=2. For each value of , use a graphing utility to find all values of 0 such that |f(x) 2| whenever 0 |x 3| . a. =12 b. =14 c. For any 0, make a conjecture about the value of that satisfies the preceding inequality.Limit proofs Use the precise definition of a limit to prove the following limits. 19. limx1(8x+5)=13Limit proofs Use the precise definition of a limit to prove the following limits. 20. limx3(2x+8)=2Limit proofs Use the precise definition of a limit to prove the following limits. 21. limx4x216x4=8 (Hint: Factor and simplify.)Limit proofs Use the precise definition of a limit to prove the following limits. 22. limx3x27x+12x3=1Limit proofs Use the precise definition of a limit to prove the following limits. Specify a relationship between and that guarantees the limit exists. 23.limx0|x|=0Limit proofs Use the precise definition of a limit to prove the following limits. Specify a relationship between and that guarantees the limit exists. 24.limx0|5x|=0Limit proofs Use the precise definition of a limit to prove the following limits. Specify a relationship between and that guarantees the limit exists. 25.limx7f(x)=9, where f(x)={3x12ifx7x+2ifx7Limit proofs Use the precise definition of a limit to prove the following limits. Specify a relationship between and that guarantees the limit exists. 26.limx5f(x)=4, where f(x)={2x6ifx54x+24ifx5Limit proofs Use the precise definition of a limit to prove the following limits. 23. limx0x2=0 (Hint: Use the identity x2=x.)Limit proofs Use the precise definition of a limit to prove the following limits. 24. limx3(x3)2=0 (Hint: Use the identity x2=x.)Limit proofs Use the precise definition of a limit to prove the following limits. Specify a relationship between and that guarantees the limit exists. 29.limx2(x23x)=10Limit proofs Use the precise definition of a limit to prove the following limits. Specify a relationship between and that guarantees the limit exists. 30.limx4(2x24x+1)=17Limit proofs Use the precise definition of a limit to prove the following limits. Specify a relationship between and that guarantees the limit exists. 31.limx3|2x|=6 (Hint: Use the inequality ab|ab|, which holds for all constants a and b (see Exercise 74).)Limit proofs Use the precise definition of a limit to prove the following limits. Specify a relationship between and that guarantees the limit exists. 32.limx25x=5 (Hint: The factorization x25=(x5)(x+5) implies that x5=x25x+5.)Challenging limit proofs Use the definition of a limit to prove the following results. 35. limx31x=13 (Hint: As x 3, eventually the distance between x and 3 is less than 1. Start by assuming |x 3| 1 and show 1x12.)Challenging limit proofs Use the definition of a limit to prove the following results. 36. limx4x4x2=4 (Him: Multiply the numerator and denominator by x+2.)Challenging limit proofs Use the definition of a limit to prove the following results. 37. limx1/101x=10 (Hint: To find , you need to bound x away from 0. So let |x110|120.)Limit proofs Use the precise definition of a limit to prove the following limits. Specify a relationship between and that guarantees the limit exists. 36.limx0sin1x=0Limit proofs Use the precise definition of a limit to prove the following limits. Specify a relationship between and that guarantees the limit exists. 37.limx0(x2+x4)=0 (Hint: You may use the fact that if |x|c, then x2c2.)Limit proofs Use the precise definition of a limit to prove the following limits. Specify a relationship between and that guarantees the limit exists. 38.limxab=b, for any constants a and bLimit proofs Use the precise definition of a limit to prove the following limits. Specify a relationship between and that guarantees the limit exists. 39.limxa(mx+b)=ma+b, for any constants a, b, and mLimit proofs Use the precise definition of a limit to prove the following limits. Specify a relationship between and that guarantees the limit exists. 40.limx3x3=27Limit proofs Use the precise definition of a limit to prove the following limits. Specify a relationship between and that guarantees the limit exists. 41.limx1x4=1Challenging limit proofs Use the definition of a limit to prove the following results. 38. limx51x2=125Proof of Limit Law 2 Suppose limxaf(x)=L and limxag(x)=M. Prove that limxa(f(x)g(x))=LM.Proof of Limit Law 3 Suppose limxaf(x)=L. Prove that limxa[cf(x)]=cL, where c is a constant.Limit proofs for infinite limits Use the precise definition of infinite limits to prove the following limits. 29. limx41(x4)2=Limit proofs for infinite limits Use the precise definition of infinite limits to prove the following limits. 30. limx11(x+1)4=Limit proofs for infinite limits Use the precise definition of infinite limits to prove the following limits. 31. limx0(1x2+1)=Limit proofs for infinite limits Use the precise definition of infinite limits to prove the following limits. 32. limx0(1x4sinx)=Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers and assume limxaf(x)=L. a. For a given 0, there is one value of 0 for which |f(x) L| whenever 0 |x a| . b. The limit limxaf(x)=L means that given an arbitrary 0, we can always find an 0 such that |f(x) L| whenever 0 |x a| . c. The limit limxaf(x)=L means that for any arbitrary 0, we can always find a 0 such that |f(x) L| whenever 0 |x a| . d. If |x a| , then a x a + .50E51E52EPrecise definitions for left- and right-sided limits Use the following definitions. Assume f exists for all x near a with x a. We say the limit of f(x) as x approaches a from the right of a is L and write limxa+f(x)=L, if for any 0 there exists 0 such that f(x)Lwhenever0xa. Assume f exists for all x near a with x a. We say the limit of f(x) as x approaches a from the left of a is L and write limxaf(x)=L, if for any 0 there exists 0 such that f(x)Lwhenever0ax. 41. One-sided limit proofs Prove the following limits for f(x)={3x4ifx02x4ifx0. a. limx0+f(x)=4 b. limx0f(x)=4 c. limx0f(x)=4Precise definitions for left- and right-sided limits Use the following definitions. Assume f exists for all x near a with x a. We say the limit of f(x) as x approaches a from the right of a is L and write limxa+f(x)=L, if for any 0 there exists 0 such that f(x)Lwhenever0xa. Assume f exists for all x near a with x a. We say the limit of f(x) as x approaches a from the left of a is L and write limxaf(x)=L, if for any 0 there exists 0 such that f(x)Lwhenever0ax. 42. Determining values of from a graph The function f in the figure satisfies limx2+f(x)=0 and limx2f(x)=1. Determine all values of 0 that satisfy each statement a. |f(x) 0| 2 whenever 0 x 2 b. |f(x) 0| 1 whenever 0 x 2 c. |f(x) 1| 2 whenever 0 2 x d. |f(x) 1| 1 whenever 0 2 x55EThe relationship between one-sided and two-sided limits Prove the following statements to establish the fact that limxaf(x)=L if and only if limxaf(x)=L and limxa+f(x)=L. a. If limxaf(x)=L and limxa+f(x)=L, then limxaf(x)=L. b. If limxaf(x)=L, then limxaf(x)=L and limxa+f(x)=L.Definition of one-sided infinite limits We write limxa+f(x)= if for any negative number N there exists 0 such that f(x)Nwhenever0xa. a. Write an analogous formal definition for limxa+f(x)=. b. Write an analogous formal definition for limxaf(x)=. c. Write an analogous formal definition for limxaf(x)=.One-sided infinite limits Use the definitions given in Exercise 45 to prove the following infinite limits. 46. limx1+11x= 45. Definition of one-sided infinite limits We write limxa+f(x)= if for any negative number N there exists 0 such that f(x)Nwhenever0xa. a. Write an analogous formal definition for limxa+f(x)=. b. Write an analogous formal definition for limxaf(x)=. c. Write an analogous formal definition for limxaf(x)=.59EDefinition of an infinite limit We write limxaf(x)= if for any negative number M there exists 0 such that f(x)Mwhenever0xa. Use this definition to prove the following statements. 48. limx12(x1)2=61ESuppose limxaf(x)=. Prove that limxa(f(x)+c)= for any constant c.Suppose limxaf(x)= and limxa(x)=. Prove that limxa(f(x)+g(x))=.Definition of a limit at infinity The limit at infinity limxf(x)=L means that for any 0 there exists N 0 such that f(x)LwheneverxN. Use this definition to prove the following statements. 50. limx10x=0Definition of a limit at infinity The limit at infinity limxf(x)=L means that for any 0 there exists N 0 such that f(x)LwheneverxN. Use this definition to prove the following statements. 51. limx2x+1x=2Definition of infinite limits at infinity We write limxf(x)= if for any positive number M there is a corresponding N 0 such that f(x)MwheneverxN. Use this definition to prove the following statements. 52. limxx100=Definition of infinite limits at infinity We write limxf(x)= if for any positive number M there is a corresponding N 0 such that f(x)MwheneverxN. Use this definition to prove the following statements. 53. limxx2+xx=68E69EProving that limxaf(x)L Use the following definition for the nonexistence of a limit. Assume f is defined for all values of x near a, except possibly at a. We write limxaf(x)L if for some 0, there is no value of 0 satisfying the condition f(x)Lwhenever0xa. 56. For the following function, note that limx2f(x)3. Find all values of 0 for which the preceding condition for nonexistence is satisfied.71EProving that limxaf(x)L Use the following definition for the nonexistence of a limit. Assume f is defined for all values of x near a, except possibly at a. We write limxaf(x)L if for some 0, there is no value of 0 satisfying the condition f(x)Lwhenever0xa. 58. Let f(x)={0ifxisrational1ifxisirrational. Prove that limxaf(x) does not exist for any value of a. (Hint: Assume limxaf(x)=L for some values of a and L and let =12.)73EShow that ab|ab| for all constants a and b (Hint Write |a|=|(ab)+b| and apply the triangle inequality to |(ab)+b|.)Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The rational function x1x21 has vertical asymptotes at x = 1 and x = 1. b. Numerical or graphical methods always produce good estimates of limxaf(x). c. The value of limxaf(x), if it exists, is found by calculating f(a). d. If limxaf(x)= or limxaf(x)=, then limxaf(x) does not exist. e. If limxaf(x) does not exist, then either limxaf(x)= or limxaf(x)=. f. The line y = 2x + 1 is a slant asymptote of the function f(x)=2x2+xx3. g. If a function is continuous on the intervals (a, b) and [b, c), where a b c, then the function is also continuous on (a, c). h. If limxaf(x) can be calculated by direct substitution, then f is continuous at x = a.The height above the ground of a stone thrown upwards is given by s(t), where t is measured in seconds. After 1 second, the height of the stone is 48 feet above the ground, and after 1.5 seconds, the height of the stone is 60 feet above the ground Evaluate s(1) and s(1.5), and then find the average velocity of the stone over the time interval [1, 1.5].A baseball is thrown upwards into the air; its distance above the ground after t seconds is given by s(t)=16t2+60t+6. Make a table of average velocities to make a conjecture about the instantaneous velocity of the baseball at t = 1.5 seconds after it was thrown into the air.Estimating limits graphically Use the graph of f in the figure to find the following values, or state that they do not exist. a. f(1) b. limx1f(x) c. limx1+f(x) d. limx1f(x) e. f(1) f. limx1f(x) g. limx2f(x) h. limx3f(x) i. limx3+f(x) j. limx3f(x)Points of discontinuity Use the graph of f in the figure to determine the values of x in the interval (3, 5) at which f fails to be continuous. Justify your answers using the continuity checklist.Computing a limit graphically and analytically a. Graph y=sin2sin with a graphing utility. Comment on any inaccuracies in the graph and then sketch an accurate graph of the function. b. Estimate lim0sin2sin using the graph in part (a). c. Verify your answer to part (b) by finding the value of lim0sin2sin analytically using the trigonometric identity sin 2 = 2 sin cos .Computing a limit numerically and analytically a. Estimate limx/4cos2xcosxsinx by making a table of values of cos2xcosxsinx for values of x approaching /4. Round your estimate to four digits. b. Use analytic methods to find the value of limx/4cos2xcosxsinx.Snowboard rental Suppose the rental cost for a snowboard is 25 for the first day (or any part of the first day) plus 15 for each additional day (or any part of a day). a. Graph the function c = f(t) that gives the cost of renting a snowboard for t days, for 0 t 5. b. Evaluate limt2.9f(t). c. Evaluate limt3f(t) and limt3+f(t). d. Interpret the meaning of the limits in part (c). e. For what values of t is f continuous? Explain.Sketching a graph Sketch the graph of a function f with all the following properties. limx2f(x)=limx2+f(x)=limx0f(x)=limx3f(x)=2limx3+f(x)=4f(3)=1Evaluating limits Determine the following limits analytically. 8. limx1000182Evaluating limits Determine the following limits analytically. 9. limx15x+612RECalculating limits Determine the following limits. 13.limh0(h+6)2+(h+6)42hCalculating limits Determine the following limits. 14.limxa(3x+1)2(3a+1)2xa, where a is constantEvaluating limits Determine the following limits analytically. 11. limx1x37x2+12x4xEvaluating limits Determine the following limits analytically. 12. limx4x37x2+12x4xEvaluating limits Determine the following limits analytically. 13. limx11x2x28x+7Evaluating limits Determine the following limits analytically. 14. limx33x+165x3Evaluating limits Determine the following limits analytically. 15. limx31x3(1x+112)Evaluating limits Determine the following limits analytically. 16. limt1/3t1/3(3t1)2Evaluating limits Determine the following limits analytically. 17. limx3x481x3Evaluating limits Determine the following limits analytically. 18. limp1p51p1Evaluating limits Determine the following limits analytically. 19. limx81x43x81Calculating limits Determine the following limits. 24.lim/2sin25sin+4sin21Evaluating limits Determine the following limits analytically. 21. limx/21sinx1x+/2One-sided limits Analyze limx1+x1x3 and limx1x1x3.Finding infinite limits Analyze the following limits. 25. limx5x7x(x5)2Finding infinite limits Analyze the following limits. 26. limx5+x5x+5Finding infinite limits Analyze the following limits. 27. limx3x4x23xFinding infinite limits Analyze the following limits. 28. limu0+u1sinuCalculating limits Determine the following limits. 31.limx1+4x34x2|x1|Calculating limits Determine the following limits. 33.limx02tanxFinding infinite limits Analyze the following limits. 29. limx02tanxCalculating limits Determine the following limits. 34.limx4x2+3x+18x4+2Limits at infinity Evaluate the following limits or state that they do not exist. 31. limx2x34x+10Limits at infinity Evaluate the following limits or state that they do not exist. 32. limxx41x5+2Calculating limits Determine the following limits. 37.limx3x+1ax2+2, where a is a positive constantCalculating limits Determine the following limits. 38.limx(x2+axx2b), where a and b are constants39RELimits at infinity Evaluate the following limits or state that they do not exist. 34. limz(e2z+2z)Limits at infinity Evaluate the following limits or state that they do not exist. 35. limx(3tan1x+2)Limits at infinity Evaluate the following limits or state that they do not exist. 33. limx(3x3+5)Calculating limits Determine the following limits. 43.limx(|x1|+x) and limx(|x1|+x)Calculating limits Determine the following limits. 44.limx(|x2|+x)x and limx(|x2|+x)x45RECalculating limits Determine the following limits. 46.limr12+er and limr12+erCalculating limits Determine the following limits. 47.limr2e4r+3e5r7e4r9e5r and limr2e4r+3e5r7e4r9e5r48RECalculating limits Determine the following limits. 49.limx(5+cos4xx2+x+1)50RECalculating limits Determine the following limits. 51.limx1xlnxApplying the Squeeze Theorem Assume the function g satisfies the inequality 1 g(x) sin2 x + 1, for x near 0. Use the Squeeze Theorem to find limx0g(x).Applying the Squeeze Theorem a. Use a graphing utility to illustrate the inequalities cosxsinxx1cosx on [1, 1]. b. Use part (a) and the Squeeze Theorem to explain why limxsinxx=1.Finding vertical asymptotes Let f(x)=x25x+6x22x. a. Analyze limx0f(x), limx0+f(x), limx2f(x), and limx2+f(x). b. Does the graph of f have any vertical asymptotes? Explain. c. Graph f and then sketch the graph with paper and pencil, correcting any errors obtained with the graphing utility.End behavior Determine the end behavior of the following functions. 37. f(x)=4x3+11x3End behavior Determine the end behavior of the following functions. 38. f(x)=x+19x2+xEnd behavior Determine the end behavior of the following functions. 39. f(x) = 1 e2xEnd behavior Determine the end behavior of the following functions. 40. f(x)=1lnx2End behavior Evaluate limxf(x) and limxf(x). 59.f(x)=6ex+203ex+4End behavior Evaluate limxf(x) and limxf(x) 60.f(x)=x+19x2+xSlant asymptotes a. Analyze limxf(x) and limxf(x) for each function. b. Determine whether f has a slant asymptote. If so, write the equation of the slant asymptote. 61.f(x)=3x2+5x+7x+1Slant asymptotes a. Analyze limxf(x) and limxf(x) for each function. b. Determine whether f has a slant asymptote. If so, write the equation of the slant asymptote. 44. f(x)=9x2+4(2x1)2Slant asymptotes a. Analyze limxf(x) and limxf(x) for each function. b. Determine whether f has a slant asymptote. If so, write the equation of the slant asymptote. 45. f(x)=1+x2x2x3x2+1Slant asymptotes a. Analyze limxf(x) and limxf(x) for each function. b. Determine whether f has a slant asymptote. If so, write the equation of the slant asymptote. 46. f(x)=x(x+2)33x24xSlant asymptotes a. Analyze limxf(x) and limxf(x) for each function. b. Determine whether f has a slant asymptote. If so, write the equation of the slant asymptote. 65.f(x)=4x3+x2+7x2x+1Finding asymptotes Find all the asymptotes of the following functions. 66.f(x)=2x2+62x2+3x2Finding asymptotes Find all the asymptotes of the following functions. 67.f(x)=1tan1xFinding asymptotes Find all the asymptotes of the following functions. 68.f(x)=2x27x2Two slant asymptotes Explain why the function f(x)=x+xex+10ex2(ex+1) has two slant asymptotes, y=12 and y=12x+5. Plot a graph of f together with its two slant asymptotes.70REContinuity at a point Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answers. 47. f(x)=1x5;a=5Continuity at a point Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answers. 48. g(x)={x216x4ifx49ifx=4;a=4Continuity at a point Use the continuity checklist to determine whether the following functions are continuous at the given value of a. 73.h(x)={2x+14ifx5x29ifx5;a=5g(x)={x35x2+6xx2ifx22ifx=2;a=2Continuity on intervals Find the intervals on which the following functions are continuous. Specify right- or left- continuity at the endpoints. 51. f(x)=x25Continuity on intervals Find the intervals on which the following functions are continuous. Specify right- or left-continuity at the endpoints. 52. g(x)=ex277REContinuity on intervals Find the intervals on which the following functions are continuous. Specify right- or left-continuity at the endpoints. 54. g(x) = cos ex79RE80RE81REIntermediate Value Theorem a. Use the Intermediate Value Theorem to show that the equation has a solution in the given interval. b. Estimate a solution to the equation in the given interval using a root finder. 82x5+7x+5=0;(1,0)x=cosx;(0,2)Suppose on a certain day the low temperature was 32 at midnight, the high temperature was 65 at noon, and then the temperature dropped to 32 the following midnight. Prove there were two times during that day, which were 12 hours apart, when the temperatures were equal (Hint: Let T(t) equal the temperature t hours after midnight and examine the function f(t)=T(t)T(t+12), for 0t12.)Antibiotic dosing The amount of an antibiotic (in mg) in the blood t hours after an intravenous line is opened is given by m(t)=100(e0.1te0.3t). a. Use the Intermediate Value Theorem to show the amount of drug is 30 mg at some time in the interval [0, 5] and again at some time in the interval [5, 15]. b. Estimate the times at which m = 30 mg. c. Is the amount of drug in the blood ever 50 mg?Limit proof Give a formal proof that limx1(5x2)=3.Limit proof Give a formal proof that limx5x225x5=10.88RE89RElimx2+4x8=0Infinite limit proof Give a formal proof that limx21(x2)4=.Limit proofs a. Assume | f(x)| L for all x near a and limxag(x)=0. Give a formal proof that limxa(f(x)g(x))=0. b. Find a function f for which limx2(f(x)(x2))0. Why doesnt this violate the result stated in (a)? c. The Heaviside function is defined as H(x)={0ifx01ifx0. Explain why limx0[xH(x)]=0.Quick Check 1
In Example 1, is the slope of the tangent line at (2, 128) greater than or less than the slope at (1, 80)?
Sketch the graph of a function f near a point a. As in Figure 3.5, draw a secant line that passes through (a, f(a)) and a neighboring point (x, f(x)) with x a. Use arrows to show how the secant lines approach the tangent line as x approaches a.Set up the calculation in Example 3 using definition (1) for the slope of the tangent line rather than definition (2). Does the calculation appear more difficult using definition (1)? Find an equation of the line tangent to the graph of f(x)=x3+4x at (1, 5).Verify that the derivative of the function f in Example 4 at the point (8, 5) is f(8)=1/4. Then find an equation of the line tangent to the graph of f at the point (8, 5). Example 4 Derivatives and Tangent Lines Let f(x)=2x+1. Compute f(2), tha deritvative of f at x = 2, and use the result, to find an equation of the line tangant to the graph of f at (2, 3).Use definition (1) (p. 127) for the slope of a tangent line to explain how slopes of secant lines approach the slope of the tangent line at a point.Explain why the slope of a secant line can be interpreted as an average rate of change.Explain why the slope of the tangent line can be interpreted as an instantaneous rate of change.Explain the relationships among the slope of a tangent line, the instantaneous rate of change, and the value of the derivative at a point.Given a function f and a point a in its domain, what does f(a) represent?The following figure shows the graph of f and a line tangent to the graph of f at x = 6. Find f(6) and f(6).An equation of the line tangent to the graph of f at the point (2, 7) is y=4x1. Find f(2) and f(2).An equation of the line tangent to the graph of g at x = 3 is y=5x+4. Find g(3) and g(3).If h(1) = 2 and h'(1) = 3, find an equation of the line tangent to the graph of h at x = 1.If f(2)=7, find an equation of the line tangent to the graph of f at the point (2, 4).Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f(x)=5x+1 at the point (1, 4).Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f(x) = 5 at P(1, 5).Velocity functions A projectile is fired vertically upward into the air; its position (in feet) above the ground after t seconds is given by the function s(t). For the following functions, use limits to determine the instantaneous velocity of the projectile at t = a seconds for the given value of a. 13.s(t)=16t2+100t;a=1Velocity functions A projectile is fired vertically upward into the air; its position (in feet) above the ground after t seconds is given by the function s(t). For the following functions, use limits to determine the instantaneous velocity of the projectile at t = a seconds for the given value of a. 14.s(t)=16t2+128t+192;a=2Equations of tangent lines by definition (1) a. Use definition (1) (p. 127) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. c. Plot the graph of f and the tangent line at P. 9. f(x) = x2 5; P(3, 4)Equations of tangent lines by definition (1) a. Use definition (1) (p. 127) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. c. Plot the graph of f and the tangent line at P. 10. f(x) = 3x2 5x + 1; P(1, 7)Equations of tangent lines by definition (1) a. Use definition (1) (p. 127) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. c. Plot the graph of f and the tangent line at P. 13. f(x)=1x;P(1,1)Equations of tangent lines by definition (1) a. Use definition (1) (p. 127) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. c. Plot the graph of f and the tangent line at P. 14. f(x)=4x2;P(1,4)Equations of tangent lines by definition (1) a. Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. c. Plot the graph of f and the tangent line at P. 19.f(x)=3x+3;P(2,3)Equations of tangent lines by definition (1) a. Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. c. Plot the graph of f and the tangent line at P. 20.f(x)=2x;P(4,1)Equations of tangent lines by definition (2) a. Use definition (2) (p. 129) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. 15. f(x) = 2x + 1; P(0, 1)Equations of tangent lines by definition (2) a. Use definition (2) (p. 129) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. 17. f(x) = 7x; P(1, 7)Equations of tangent lines by definition (2) a. Use definition (2) (p. 129) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. 16. f(x) = 3x2 4x; P(1, 1)Equations of tangent lines by definition (2) a. Use definition (2) (p. 129) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. 18. f(x) = 8 2x2; P(0, 8)