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

This preview shows pages 1–6. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern