02 - CHAPTER 2 Limits and Their Properties Section 2.1...

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C H A P T E R 2 Limits and Their Properties Section 2.1 A Preview of Calculus . . . . . . . . . . . . . . . . . . . . 71 Section 2.2 Finding Limits Graphically and Numerically . . . . . . . . 72 Section 2.3 Evaluating Limits Analytically . . . . . . . . . . . . . . . 83 Section 2.4 Continuity and One-Sided Limits . . . . . . . . . . . . . . 94 Section 2.5 Infinite Limits . . . . . . . . . . . . . . . . . . . . . . . 105 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
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C H A P T E R 2 Limits and Their Properties Section 2.1 A Preview of Calculus 71 1. Precalculus: 20 ft sec 15 seconds 300 feet 2. Calculus: velocity is not constant Distance 20 ft sec 15 seconds 300 feet 3. Calculus required: slope of tangent line at is rate of change, and equals about 0.16. x 2 4. Precalculus: rate of change slope 0.08 5. Precalculus: sq. units Area 1 2 bh 1 2 5 3 15 2 6. Calculus required: 5 sq. units 2 2.5 Area bh 7. (a) (b) (c) At the slope is 2. You can improve your approximation of the slope at by considering values very close to 1. x - x 1 P 1, 3 m 3 0.5 2.5 x 0.5: m 3 1.5 1.5 x 1.5: m 3 2 1 x 2: x 1 x 1 3 x x 1 3 x , slope m 4 x x 2 3 x 1 1 2 3 3 4 P y x f x 4 x x 2 9. (a) (b) You could improve the approximation by using more rectangles. Area 1 2 5 5 1.5 5 2 5 2.5 5 3 5 3.5 5 4 5 4.5 9.145 Area 5 5 2 5 3 5 4 10.417 8. (a) (b) (c) At the slope is You can improve your approximation of the slope at by considering values very close to 4. x - x 4 1 4 2 1 4 0.25. P 4, 2 m 1 5 2 0.2361 x 5: m 1 3 2 0.2679 x 3: m 1 1 2 1 3 x 1: x 4 x 2 x 2 x 2 1 x 2 , slope m x 2 x 4 x y 1 2 3 4 5 2 P (4, 2) f x x
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72 Chapter 2 Limits and Their Properties 10. (a) For the figure on the left, each rectangle has width For the figure on the right, each rectangle has width (b) You could obtain a more accurate approximation by using more rectangles. You will learn later that the exact area is 2. 3 2 6 1.9541 6 1 2 3 2 1 3 2 1 2 Area 6 sin 6 sin 3 sin 2 sin 2 3 5 6 sin 6 . 2 1 4 1.8961 4 2 2 1 2 2 Area 4 sin 4 sin 2 sin 3 4 sin 4 . 11. (a) (b) (c) Increase the number of line segments. 2.693 1.302 1.083 1.031 6.11 D 2 1 5 2 2 1 5 2 5 3 2 1 5 3 5 4 2 1 5 4 1 2 D 1 5 1 2 1 5 2 16 16 5.66 Section 2.2 Finding Limits Graphically and Numerically 1. Actual limit is 1 3 . lim x 2 x 2 x 2 x 2 0.3333 1.9 1.99 1.999 2.001 2.01 2.1 0.3448 0.3344 0.3334 0.3332 0.3322 0.3226 f x x 2. Actual limit is 1 4 . lim x 2 x 2 x 2 4 0.25 x 1.9 1.99 1.999 2.001 2.01 2.1 0.2564 0.2506 0.2501 0.2499 0.2494 0.2439 f x 3. Actual limit is 1 16 . lim x 3 1 x 1 1 4 x 3 0.0625 x 2.9 2.99 2.999 3.001 3.01 3.1 0.0610 0.0623 0.0625 0.0625 0.0627 0.0641 f x
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Section 2.2 Finding Limits Graphically and Numerically 73 4. 1 4 . Actual limit is lim x 3 1 x 2 x 3 0.25 x 3.1 3.01 3.001 2.999 2.99 2.9 0.2516 0.2502 0.2500 0.2500 0.2498 0.2485 f x 5. (Actual limit is 1.) (Make sure you use radian mode.) lim x 0 sin x x 1.0000 x 0.1 0.01 0.001 0.001 0.01 0.1 0.9983 0.99998 1.0000 1.0000 0.99998 0.9983 f x 6. (Actual limit is 0.) (Make sure you use radian mode.) lim x 0 cos x 1 x 0.0000 x 0.1 0.01 0.001 0.001 0.01 0.1 0.0500 0.0050 0.0005 0.0005 0.0050 0.0500 f x 7. lim x 0 e x 1 x 1 x 0.1 0.01 0.001 0.001 0.01 0.1 0.9516 0.9950 0.9995 1.0005 1.0050 1.0517 f x 8. does not exist. lim x 0 4 1 e 1 x x 0.1 0.01 0.001 0.001 0.01 0.1 3.99982 4 4 0 0 0.00018 f x 9. lim x 0 ln x 1 x 1 x 0.1 0.01 0.001 0.001 0.01 0.1 1.0536 1.0050 1.0005 0.9995 0.9950 0.9531 f x 10. lim x 2 ln x ln 2 x 2 1 2 x 1.9 1.99 1.999 2.001 2.01 2.1 0.5129 0.5013 0.5001 0.4999 0.4988 0.4879 f x
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25. exists for all values of c 4. lim x c f x 1 2 1 2 3 4 5 1 2 1 2 3 4 5 6 y x f 26.
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