Chapter 2 Limits 2.2 The Limit of a Function. 30. [T] Complete the following table for the function. Round your solutions to four decimal places.
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1 OpenSta Calculus Chapter Limits. The Limit of a Function Section Eercises For the following eercises, consider the function f [T] Complete the following table for the function. Round your solutions to four decimal places. f X f 0.9 a. 1.1 e b f c g d h. Answer: a ; b ; c ; d ; e..1000; f..0100; g..0010; h What do your results in the preceding eercise indicate about the two-sided it f? Eplain your response. Answer: 1 f 1 does not eist because f f. 1 1 For the following eercises, consider the function f 1 1/. 3. [T]Make a table showing the values of f for 0.01, 0.001, , and for 0.01, 0.001, , Round your solutions to five decimal places. f f 0.01 a e b f c g d h. Answer: a..7300; b ; c..7184; d ; e ; f..7169; g ; h What does the table of values in the preceding eercise indicate about the function ( ) = ( 1+ ) 1/? f Answer:
2 OpenSta Calculus 34. To which mathematical constant does the it in the preceding eercise appear to be getting closer? Answer: e.7183 In the following eercises, use the given values to set up a table to evaluate the its. Round your solutions to eight decimal places. 35. sin [T] ; ±0.1, ±0.01, ±0.001, ± sin sin 0.1 a. 0.1 e b f c g d h. Answer: a ; b ; c ; d ; e ; f. sin ; g ; h ; sin3 [T] ; ±0.1, ±0.01, ±0.001, ± X sin 3 sin a. 0.1 e b f c g d h. Answer: a ; b ; c ; d ; e ; f. sin ; g ; h ; Use the preceding two eercises to conjecture (guess) the value of the following it: sin a for a, a positive real value. 0 sin a Answer: a 0
3 OpenSta Calculus [T] In the following eercises, set up a table of values to find the indicated it. Round to eight digits a..1 e b..01 f c..001 g d h. Answer: a ; b ; c ; d ; e ; f ; g ; h ; a. 1.1 e b f c g d h. Answer: a ; b ; c ; d ; e ; f ; g ; h ; e 1/ 5 e e a. 0.1 e b f c g d h. Answer: a ; b ; c ; d ; e ; f ; g ; h ; Limit does not eist. 41. z z0 z z 1 z 3 z z 1 z a. 0.1 e b f c g d h. z z z 1 z 3
4 OpenSta Calculus Answer: a ; b ; c. 333,777.93; d. 33,337,778; e ; f. z ; g. 33,889.04; h. 33,38,889 0 z z 3 4. cost t0 t t cos t t 0.1 a b c d. Answer: a ; b ; c ; d. 10, ; cost t t a..1 e b..01 f c..001 g d h. Answer: a ; b ; c ; d ; e ; f ; g ; h ; ] 4 8 [T] In the following eercises, set up a table of values and round to eight significant digits. Based on the table of values, make a guess about what the it is. Then, use a calculator to graph the function and determine the it. Was the conjecture correct? If not, why does the method of tables fail? 44. sin 0 θ sin θ sin 0.1 a. 0.1 e b f c g d h.
5 OpenSta Calculus Answer: a ; b ; c ; d ; e ; f ; g ; h ; Guess: sin 0, actual: DNE cos 0 a 1 cos 0.1 a b c d. Answer: a ; b ; c ; d. 10, ; Guess: 1 cos, actual: DNE 0
6 OpenSta Calculus y f shown here. y f are true and which are false? Eplain why a In the following eercises, consider the graph of the function Which of the statements about statement is false. 46. f 0 10 Answer: True 47. f 3 Answer: False; 48. f f 8 8 Answer: False; 8 f whereas f 6 f f 5 6 Answer: False; f 6 DNE since 6 and f f 5. 6
7 OpenSta Calculus In the following eercises, use the following graph of the function values, if possible. Estimate when necessary. y f to find the 50. f 1 Answer: f 1 Answer: 5. f 1 Answer: DNE 53. f Answer: f Answer: 1
8 OpenSta Calculus In the following eercises, use the graph of the function values, if possible. Estimate when necessary. y f shown here to find the 55. f 0 Answer: f 0 Answer: f 0 Answer: DNE 58. f In the following eercises, use the graph of the function values, if possible. Estimate when necessary. y f shown here to find the 59. f
9 OpenSta Calculus 60. f Answer: 61. f Answer: DNE 6. f Answer: 63. f Answer: 64. f Answer: DNE In the following eercises, use the graph of the function values, if possible. Estimate when necessary. y g shown here to find the 65. g 0 Answer: g g 0 Answer: DNE
10 OpenSta Calculus In the following eercises, use the graph of the function values, if possible. Estimate when necessary. y h shown here to find the 68. h h h 0 In the following eercises, use the graph of the function values, if possible. Estimate when necessary. y f shown here to find the 71. f 0 Answer:
11 OpenSta Calculus 7. f f 0 Answer: DNE 74. f 1 Answer: f In the following eercises, sketch the graph of a function with the given properties. 76. f 1, f 3 4 Answer: Answers may vary., f 6, 4 is not defined f 0, f f 1, 1 f, f f 0, 0 f 0 1, Answer: Answers may vary.
12 OpenSta Calculus 78. f, f 3 Answer: Answers may vary., 3 f,, f 0 f f, f Answer: Answers may vary.,, f f f 0, f 1 f, 1, f 0 Answer: Answers may vary. 1 f, f 0 1, f 1,
13 OpenSta Calculus 81. Shock waves arise in many physical applications, ranging from supernovas to detonation waves. A graph of the density of a shock wave with respect to distance,, is shown here. We are mainly interested in the location of the front of the shock, labeled SF in the diagram. a. Evaluate. SF b. Evaluate. SF c. Evaluate SF. Eplain the physical meanings behind your answers. Answer: a. b. 1 c. DNE unless 1. As you approach SF from the right, you are in the high-density area of the shock. When you approach from the left, you have not eperienced the shock yet and are at a lower density. 8. A track coach uses a camera with a fast shutter to estimate the position of a runner with respect to time. A table of the values of position of the athlete versus time is given here, where is the position in meters of the runner and t is time in seconds. What is t? What does it mean physically? t t (sec) (m) Answer: t 6.5. As the runner approaches sec, her position is roughly 6.5 m. t This file is copyright 016, Rice University. All Rights Reserved.
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