Preview Calculus Early Transcendentals by William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz (2018)

Preview Calculus Early Transcendentals by William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz (2018) Preview Calculus Early Transcendentals by William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz (2018) Preview Calculus Early Transcendentals by William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz (2018) Preview Calculus Early Transcendentals by William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz (2018)

BRIGGS COCHRAN GILLETT SCHULZ Get the Most Out of MyLab Math S Interactive practice with immediate feedback and just-in-time learning aids S Multimedia learning resources S Complete eText, accessible anywhere with the Pearson eText app MyLab Math is available for this textbook To learn more, visit pearson.com/mylab/math CALCULUS S Personalized learning experiences EARLY TRANSCENDENTALS MyLabTM Math is the world’s leading online tutorial and assessment program designed to help you learn and succeed in your mathematics course MyLab Math courses provide you with: T H I R D E D I T I O N T H I R D E D I T I O N CALCULUS EARLY TRANSCENDENTALS www.pearson.com BRIGGS  • COCHRAN • GILLETT • SCHULZ MyLab Math for Calculus: Early Transcendentals, 3e (access code required) Used by over million students a year, MyLab™ Math is the world’s leading online program for teaching and learning mathematics MyLab Math for Calculus: Early Transcendentals, 3e delivers text-specific assessment, tutorials, and multimedia resources that provide engaging and personalized experiences, so learning can happen in any environment and course format eBook with Interactive Figures This groundbreaking eBook includes approximately 700 figures that can be manipulated to provide a deeper geometric understanding of key concepts Plus, all-new Enhanced Interactive Figures incorporate functionality from several standard Interactive Figures, making each one mathematically richer and ideal for in-class demonstrations Questions that Deepen Understanding MyLab Math includes a variety of question types designed to help students succeed in the course In Setup & Solve questions, students show how they set up a problem as well as the solution, better mirroring what is required on tests Additional Conceptual Questions were written by faculty at Cornell University to support deeper, theoretical understanding of the key concepts in calculus ALL NEW! Instructional Videos For each section of the text, there is a newly recorded full-lecture video Many videos make use of Interactive Figures to enhance understanding of concepts To make them easier to navigate, each lecture video is segmented into parts (Introduction, Example, or Summary) The videos are assignable within MyLab Math, and a Guide to Video-Based Assignments shows which MyLab Math exercises correspond to each video pearson.com/mylab/math A00_BRIG3644_03_SE_FEP2.indd 27/10/17 3:54 PM ALGEBRA Exponents and Radicals xa x a xa a-b -a a b ab b = x    x =    1x = x    a = y xa ya xb n n m n n n n n n n m>n m = 2x   x = 2x = 12x2    2xy = 2x2y   2x>y = 2x> 2y x ax b = x a + b    x 1>n Factoring Formulas Binomials 2 1a { b2 = a2 { 2ab + b2 1a { b2 = a3 { 3a2b + 3ab2 { b3 a - b = 1a - b21a + b2 a + b does not factor over real numbers a3 - b3 = 1a - b21a2 + ab + b22 a3 + b3 = 1a + b21a2 - ab + b22 n n n-1 n-2 n-3 a - b = 1a - b21a + a b + a b + g + abn - + bn - 12 Binomial Theorem Quadratic Formula n n n 1a + b2 n = an + a b an - 1b + a b an - 2b2 + g + a b abn - + bn, n - n1n - 121n - 22 g1n - k + 12 n n! where a b = = k k1k - 121k - 22 g3 # # k!1n - k2! The solutions of ax + bx + c = are x = -b { 2b2 - 4ac   2a GEOMETRY Triangle Parallelogram Trapezoid Sector Circle a h h r h b b A bh A5 Cylinder b bh A5 Cone s u (a b)h 2 r u s ru (u in radians) A pr A5 C 2pr Equations of Lines and Circles Sphere r y2 - y1 slope of line through 1x1, y12 and 1x2, y22 x2 - x1 y - y1 = m1x - x12 point–slope form of line through 1x1, y12 with slope m y = mx + b slope–intercept form of line with slope m and y-intercept 10, b2 1x - h2 + 1y - k2 = r circle of radius r with center 1h, k2 m = h r , h r r 2h V r r h S r, (lateral surface area) V S rh (lateral surface area) V S r r2 y (x1, y1) y y2 y1 m5 x 2x (0, b) r y mx b (h, k) O (x2, y2) O x x (x h)2 (y k)2 r TRIGONOMETRY o yp h u adjacent y opposite se u ten adj opp opp cos u = sin u = tan u = hyp hyp adj hyp hyp adj sec u = csc u = cot u = opp opp adj (x, y) r x r y sin u = r y tan u = x cos u = u x r x r csc u = y x cot u = y sec u = (Continued) A00_BRIG3644_03_SE_FEP.indd 08/09/17 2:20 PM /2 /3 /6 /4 /4 ( /6 45 /3 90 15 tan u = ( 22 , 22 ) 60 ) Reciprocal Identities ( 12 , 23 ) , 2 13 ( , (0, 1) ) 2, 12 ( ( 2 ) , 2 In general, cos u = x; sin u = y y 180 /4 /3 /2 24 , 270 ( ) /4 2 ) 11 /3 , (1, 0) /6 ( , ( 12 , (0, 1) ( 22 , ) tan 2u = cos u csc u = sin u sin2 u + cos2 u = tan2 u + = sec2 u + cot2 u = csc2 u Sign Identities ) ) Half-Angle Identities cos 2u = cos2 u - sin2 u = cos2 u - = - sin2 u tan u - tan2 u 2 Double-Angle Identities sin 2u = sin u cos u sec u = sin 1-u2 = -sin u cos 1-u2 = cos u tan 1-u2 = -tan u csc 1-u2 = -csc u sec 1-u2 = sec u cot 1-u2 = -cot u 33 cos u sin u 22 ) 21 ( /6 31 30 ( , ( 1, 0) 0 radians 360 x cot u = Pythagorean Identities ) (x, y) 30 sin u cos u cos2 u = + cos 2u sin2 u = - cos 2u Addition Formulas sin 1a + b2 = sin a cos b + cos a sin b cos 1a + b2 = cos a cos b - sin a sin b tan a + tan b tan 1a + b2 = - tan a tan b Law of Sines a b sin b sin g sin a = = a c b sin 1a - b2 = sin a cos b - cos a sin b cos 1a - b2 = cos a cos b + sin a sin b tan a - tan b tan 1a - b2 = + tan a tan b Law of Cosines b g a2 = b2 + c2 - 2bc cos a a c Graphs of Trigonometric Functions and Their Inverses y Range of tan p p is , ( ) x y y tan x y p sin y x y y x y sin x cos x y x Range of cos x Domain of cos p 1 x y p 1 p x [ 1, 1] y 1 Range of sin x Domain of sin p [ ] p p , 2 Restricted domain of sin x Range of sin x A00_BRIG3644_03_SE_FEP.indd x cos x x p [ 1, 1] 1 p tan x x 1 p y x [0, ] Restricted domain of cos x Range of cos x Restricted domain of tan x p p is , ( ) 08/09/17 2:20 PM Calculus E A R LY T R A N S C E N D E N TA L S Third Edition WILLIAM BRIGGS University of Colorado, Denver LYLE COCHRAN Whitworth University BERNARD GILLETT University of Colorado, Boulder ERIC SCHULZ Walla Walla Community College A01_BRIG3644_03_SE_FM_i-xxii.indd 07/11/17 5:37 PM Director, Portfolio Management: Executive Editor: Content Producer: Managing Producer: Producer: Manager, Courseware QA: Manager, Content Development: Product Marketing Manager: Field Marketing Manager: Marketing Assistants: Senior Author Support/Technology Specialist: Manager, Rights and Permissions: Manufacturing Buyer: Text and Cover Design, Production Coordination, Composition: Illustrations: Cover Image: Deirdre Lynch Jeff Weidenaar Rachel S Reeve Scott Disanno Stephanie Green Mary Durnwald Kristina Evans Emily Ockay Evan St Cyr Shannon McCormack, Erin Rush Joe Vetere Gina Cheselka Carol Melville, LSC Communications Cenveo® Publisher Services Network Graphics, Cenveođ Publisher Services Nutexzles/Moment/Getty Images Copyright â 2019, 2015, 2011 by Pearson Education, Inc All Rights Reserved Printed in the United States of America This publication is protected by copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise For information regarding permissions, request forms, and the appropriate contacts within the Pearson Education Global Rights & Permissions department, please visit www.pearsoned.com/permissions/ Attributions of third party content appear on page xxii, which constitutes an extension of this copyright page PEARSON, ALWAYS LEARNING, and MYLAB are exclusive trademarks owned by Pearson Education, Inc or its affiliates in the U.S and/or other countries Unless otherwise indicated herein, any third-party trademarks that may appear in this work are the property of their respective owners and any ­references to third-party trademarks, logos or other trade dress are for demonstrative or descriptive purposes only Such references are not intended to imply any sponsorship, endorsement, authorization, or promotion of Pearson’s products by the owners of such marks, or any relationship between the owner and Pearson Education, Inc or its affiliates, authors, licensees or distributors Library of Congress Cataloging-in-Publication Data Names: Briggs, William L., author | Cochran, Lyle, author | Gillett, Bernard, author | Schulz, Eric P., author Title: Calculus Early transcendentals Description: Third edition / William Briggs, University of Colorado, Denver, Lyle Cochran, Whitworth University,   Bernard Gillett, University of Colorado, Boulder, Eric Schulz, Walla Walla Community College | New York,   NY : Pearson, [2019] | Includes index Identifiers: LCCN 2017046414 | ISBN 9780134763644 (hardcover) | ISBN 0134763645 (hardcover) Subjects: LCSH: Calculus—Textbooks Classification: LCC QA303.2 B75 2019 | DDC 515—dc23 LC record available at https://lccn.loc.gov/2017046414 1 17 Instructor’s Edition ISBN 13: 978-0-13-476684-3 ISBN 10: 0-13-476684-9 Student Edition ISBN 13: 978-0-13-476364-4 ISBN 10: 0-13-476364-5 A01_BRIG3644_03_SE_FM_i-xxii.indd 20/11/17 11:31 AM For Julie, Susan, Sally, Sue, Katie, Jeremy, Elise, Mary, Claire, Katie, Chris, and Annie, whose support, patience, and encouragement made this book possible A01_BRIG3644_03_SE_FM_i-xxii.indd 07/11/17 5:37 PM Contents Preface ix Credits xxii Functions 1.1 Review of Functions  1.2 Representing Functions  13 1.3 Inverse, Exponential, and Logarithmic Functions  27 1.4 Trigonometric Functions and Their Inverses  39 Review Exercises  51 Limits 2.1 The Idea of Limits  56 2.2 Definitions of Limits  63 2.3 Techniques for Computing Limits  71 2.4 Infinite Limits  83 2.5 Limits at Infinity  91 56 2.6 Continuity 103 2.7 Precise Definitions of Limits  116 Review Exercises  128 Derivatives 3.1 Introducing the Derivative  131 3.2 The Derivative as a Function  140 3.3 Rules of Differentiation  152 3.4 The Product and Quotient Rules  163 3.5 Derivatives of Trigonometric Functions  171 3.6 Derivatives as Rates of Change  178 3.7 The Chain Rule  191 3.8 Implicit Differentiation  201 3.9 Derivatives of Logarithmic and Exponential Functions  208 131 iv A01_BRIG3644_03_SE_FM_i-xxii.indd 07/11/17 5:38 PM Contents v 3.10 Derivatives of Inverse Trigonometric Functions  218 3.11 Related Rates  227 Review Exercises  236 Applications of the Derivative 4.1 Maxima and Minima  241 4.2 Mean Value Theorem  250 4.3 What Derivatives Tell Us  257 4.4 Graphing Functions  271 4.5 Optimization Problems  280 4.6 Linear Approximation and Differentials  292 4.7 L’Hôpital’s Rule  301 4.8 Newton’s Method  312 241 4.9 Antiderivatives 321 Review Exercises  334 Integration 5.1 Approximating Areas under Curves  338 5.2 Definite Integrals  353 5.3 Fundamental Theorem of Calculus  367 5.4 Working with Integrals  381 5.5 Substitution Rule  388 338 Review Exercises  398 Applications of Integration 6.1 Velocity and Net Change  403 6.2 Regions Between Curves  416 6.3 Volume by Slicing  425 6.4 Volume by Shells  439 6.5 Length of Curves  451 6.6 Surface Area  457 6.7 Physical Applications  465 403 Review Exercises  478 Logarithmic, Exponential, and Hyperbolic Functions 7.1 Logarithmic and Exponential Functions Revisited  483 7.2 Exponential Models  492 7.3 Hyperbolic Functions  502 483 Review Exercises  518 A01_BRIG3644_03_SE_FM_i-xxii.indd 07/11/17 5:38 PM vi Contents Integration Techniques 8.1 Basic Approaches  520 8.2 Integration by Parts  525 8.3 Trigonometric Integrals  532 8.4 Trigonometric Substitutions  538 8.5 Partial Fractions  546 8.6 Integration Strategies  556 8.7 Other Methods of Integration  562 8.8 Numerical Integration  567 8.9 Improper Integrals  582 520 Review Exercises  593 Differential Equations 9.1 Basic Ideas  597 9.2 Direction Fields and Euler’s Method  606 9.3 Separable Differential Equations  614 9.4 Special First-Order Linear Differential Equations  620 9.5 Modeling with Differential Equations  627 597 Review Exercises  636 10 Sequences and Infinite Series 639 10.1 An Overview  639 10.2 Sequences 650 10.3 Infinite Series  662 10.4 The Divergence and Integral Tests  671 10.5 Comparison Tests  683 10.6 Alternating Series  688 10.7 The Ratio and Root Tests  696 10.8 Choosing a Convergence Test  700 Review Exercises  704 11 Power Series 708 11.1 Approximating Functions with Polynomials  708 11.2 Properties of Power Series  722 11.3 Taylor Series  731 11.4 Working with Taylor Series  742 Review Exercises  750 A01_BRIG3644_03_SE_FM_i-xxii.indd 07/11/17 5:38 PM 116 Chapter 2  •  Limits 2.7  Precise Definitions of Limits The limit definitions already encountered in this chapter are adequate for most elementary limits However, some of the terminology used, such as sufficiently close and arbitrarily large, needs clarification The goal of this section is to give limits a solid mathematical foundation by transforming the previous limit definitions into precise mathematical statements ➤ The phrase for all x near a means for all Moving Toward a Precise Definition x in an open interval containing a ➤ The Greek letters d (delta) and e (epsilon) represent small positive numbers in the discussion of limits xSa equal) to a This limit definition is made precise by observing that the distance between ƒ1x2 and L is ƒ1x2 - L and that the distance between x and a is x - a Therefore, we write lim ƒ1x2 = L if we can make ƒ1x2 - L arbitrarily small for any x, distinct from a, with ➤ The two conditions x - a d y Assume the function ƒ is defined for all x near a, except possibly at a Recall that lim ƒ1x2 = L means that ƒ1x2 is arbitrarily close to L for all x sufficiently close (but not and x ≠ a are written concisely as x - a d xSa x - a sufficiently small For instance, if we want ƒ1x2 - L to be less than 0.1, then we must find a number d such that ƒ1x2 - L 0.1 whenever If, instead, we want ƒ1x2 - L to be less than 0.001, then we must find another number d such that y f (x) ƒ1x2 - L 0.001 whenever x - a d lim f (x) 5 x ¢ For the limit to exist, it must be true that for any e 0, we can always find a d such that ƒ1x2 - L e whenever x - a d 1 x - a d and x ≠ a x EXAMPLE 1  Determining values of D from a graph  Figure 2.59 shows the graph of a linear function ƒ with lim ƒ1x2 = For each value of e 0, determine a value of xS3 Figure 2.59 d satisfying the statement ƒ1x2 - e whenever x - d ➤ The founders of calculus, Isaac Newton (1642–1727) and Gottfried Leibniz (1646–1716), developed the core ideas of calculus without using a precise definition of a limit It was not until the 19th century that a rigorous definition was introduced by Augustin-Louis Cauchy (1789–1857) and later refined by Karl Weierstrass (1815–1897) u f (x) 5u , a e = 1  b.  e = SOLUTION a With e = 1, we want ƒ1x2 to be less than unit from 5, which means ƒ1x2 is between and To determine a corresponding value of d, draw the horizontal lines y = and y = (Figure 2.60a) Then sketch vertical lines passing through the points where the horizontal lines and the graph of ƒ intersect (Figure 2.60b) We see that the vertical y y 6 5 4 (a) x    Values of x such that u f (x) 5u , u f (x) 5u , 1 x , ux 3u , (b) Figure 2.60 M02_BRIG3644_03_SE_C02_056-130.indd 116 23/06/17 9:55 AM 2.7  Precise Definitions of Limits 117 lines intersect the x-axis at x = and x = Note that ƒ1x2 is less than unit from on the y-axis if x is within units of on the x-axis So for e = 1, we let d = or any smaller positive value ➤ Once an acceptable value of d is found satisfying the statement ƒ1x2 - L e whenever x - a d, any smaller positive value of d also works u f (x) 5u , 12 b With e = 12 , we want ƒ1x2 to lie within a half-unit of 5, or equivalently, ƒ1x2 must lie between 4.5 and 5.5 Proceeding as in part (a), we see that ƒ1x2 is within a half-unit of on the y-axis (Figure 2.61a) if x is less than unit from (Figure 2.61b) So for e = 12 , we let d = or any smaller positive number y y 5.5 4.5 5.5 4.5 x Values of x such that u f (x) 5u , 12 u f (x) 5u , 12 x , ux 3u , (a)    Figure 2.61 (b) Related Exercises 9–10 The idea of a limit, as illustrated in Example 1, may be described in terms of a contest between two people named Epp and Del First, Epp picks a particular number e 0; then he challenges Del to find a corresponding value of d such that y ƒ1x2 - e whenever x - d.(1) Values of x such that ) f (x) 5) , 18 51 52 ) f (x) 5) , 18 xSa To illustrate, suppose Epp chooses e = From Example 1, we know that Del will satisfy (1) by choosing d … If Epp chooses e = 12 , then (by Example 1) Del responds by letting d … If Epp lets e = 18 , then Del chooses d … 14 (Figure 2.62) In fact, there is a pattern: For any e that Epp chooses, no matter how small, Del will satisfy (1) by choosing a positive value of d satisfying d … 2e Del has discovered a mathematical relationship: If d … 2e and x - d, then ƒ1x2 - e, for any e This conversation illustrates the general procedure for proving that lim ƒ1x2 = L 32 31 , )x 3) , 14 Figure 2.62 x QUICK CHECK 1  In Example 1, find a positive number d satisfying the statement ƒ1x2 - A Precise Definition 100 whenever x - d Example dealt with a linear function, but it points the way to a precise definition of a limit for any function As shown in Figure 2.63, lim ƒ1x2 = L means that for any positive number xSa e, there is another positive number d such that ƒ1x2 - L e whenever x - a d In all limit proofs, the goal is to find a relationship between e and d that gives an admissible value of d, in terms of e only This relationship must work for any positive value of e M02_BRIG3644_03_SE_C02_056-130.indd 117 23/06/17 9:55 AM 118 Chapter 2  •  Limits lim f (x) L x y $a L1« f (x) L then u f (x) Lu , « L2« y f (x) O a2d x a x a1d If , ux au , d Figure 2.63 DEFINITION  Limit of a Function Assume ƒ1x2 is defined for all x in some open interval containing a, except possibly at a We say the limit of ƒ1 x2 as x approaches a is L, written lim ƒ1x2 = L, xSa if for any number e there is a corresponding number d such that ƒ1x2 - L e whenever x - a d EXAMPLE 2  Finding D for a given E using a graphing utility Let ƒ1x2 = x 6x + 12x - and demonstrate that lim ƒ1x2 = as follows For the given values of xS2 e, use a graphing utility to find a value of d such that ƒ1x2 - e whenever x - d ➤ The value of d in the precise definition of a limit depends only on e ➤ Definitions of the one-sided limits lim ƒ1x2 = L and lim- ƒ1x2 = L are x S a+ xSa discussed in Exercises 51–55 a e = 1  b.  e = SOLUTION a The condition ƒ1x2 - e = implies that ƒ1x2 lies between and Using a graphing utility, we graph ƒ and the lines y = and y = (Figure 2.64) These lines intersect the graph of ƒ at x = and at x = We now sketch the vertical lines x = and x = and observe that ƒ1x2 is within unit of whenever x is within unit of on the x-axis (Figure 2.64) Therefore, with e = 1, we can choose any d with d … y y f (x) u f (x) 3u , 2 x , ux 2u , Figure 2.64 M02_BRIG3644_03_SE_C02_056-130.indd 118 23/06/17 9:55 AM 2.7  Precise Definitions of Limits 119 b The condition ƒ1x2 - e = 12 implies that ƒ1x2 lies between 2.5 and 3.5 on the y-axis We now find that the lines y = 2.5 and y = 3.5 intersect the graph of ƒ at x ≈ 1.21 and x ≈ 2.79 (Figure 2.65) Observe that if x is less than 0.79 unit from on the x-axis, then ƒ1x2 is less than a half-unit from on the y-axis Therefore, with e = 12 we can choose any d with d … 0.79 y y f (x) 3.5 u f (x) 3u , 2 2.5 1.21 2.79 x , ux 2u , 0.79 Figure 2.65 This procedure could be repeated for smaller and smaller values of e For each value of e, there exists a corresponding value of d, proving that the limit exists Related Exercise 13 QUICK CHECK 2  For the function ƒ given in Example 2, estimate a value of d satisfying ƒ1x2 - 0.25 whenever x - d y EXAMPLE 3  Finding a symmetric interval  Figure 2.66 shows the graph of g with lim g1x2 = For each value of e, find the corresponding values of d that satisfy xS2 the condition g1x2 - e whenever x - d y g(x) a e = 2   b e = c For any given value of e, make a conjecture about the corresponding values of d that satisfy the limit condition The inequality x - a d means that x lies between a - d and a + d with x ≠ a We say that the interval 1a - d, a + d2 is symmetric about a because a is the midpoint of the interval Symmetric intervals are convenient, but Example demonstrates that we don’t always get symmetric intervals without a bit of extra work Figure 2.66 M02_BRIG3644_03_SE_C02_056-130.indd 119 x SOLUTION a With e = 2, we need a value of d such that g1x2 is within units of 3, which means g1x2 is between and 5, whenever x is less than d units from The horizontal lines y = and y = intersect the graph of g at x = and x = Therefore, g1x2 - if x lies in the interval 11, 62 with x ≠ (Figure 2.67a) However, we want x to lie in an interval that is symmetric about We can guarantee that g1x2 - in an interval symmetric about only if x is less than unit away from 2, on either side of (Figure 2.67b) Therefore, with e = 2, we take d = or any smaller positive number 23/06/17 9:55 AM 120 Chapter 2  •  Limits y y 5 y g(x) u g(x) 3u , y g(x) 3 1 x x Symmetric interval , ux 2u , that guarantees u g(x) 3u , Values of x such that u g(x) 3u , (a) (b)    Figure 2.67 b With e = 1, g1x2 must lie between and (Figure 2.68a) This implies that x must be within a half-unit to the left of and within units to the right of Therefore, g1x2 - provided x lies in the interval 11.5, 42 To obtain a symmetric interval about 2, we take d = 12 or any smaller positive number Then we are still guaranteed that g1x2 - when x - 12 (Figure 2.68b) y y u g(x) 3u , y g(x) 3 2 1.5 x 1.5 2.5 x Symmetric interval , ux 2u , 12 that guarantees u g(x) 3u , Values of x such that u g(x) 3u , (a) y g(x) (b)    Figure 2.68 c From parts (a) and (b), it appears that if we choose d … e>2, the limit condition is satisfied for any e Related Exercises 15–16 In Examples and 3, we showed that a limit exists by discovering a relationship between e and d that satisfies the limit condition We now generalize this procedure Limit Proofs We use the following two-step process to prove that lim ƒ1x2 = L xSa M02_BRIG3644_03_SE_C02_056-130.indd 120 23/06/17 9:55 AM 2.7  Precise Definitions of Limits 121 ➤ The first step of the limit-proving process Steps for proving that lim ƒ1 x2 = L is the preliminary work of finding a candidate for d The second step verifies that the d found in the first step actually works xSa Find D Let e be an arbitrary positive number Use the inequality ƒ1x2 - L e to find a condition of the form x - a d, where d depends only on the value of e Write a proof For any e 0, assume x - a d and use the relationship between e and d found in Step to prove that ƒ1x2 - L e EXAMPLE 4  Limit of a linear function  Prove that lim 14x - 152 = using the prexS4 cise definition of a limit SOLUTION Step 1: Find d In this case, a = and L = Assuming e is given, we use 14x - 152 - e to find an inequality of the form x - d If 14x - 152 - e, then 4x - 16 e x - e   Factor 4x - 16 e x -   Divide by and identify d = e>4 We have shown that 14x - 152 - e implies that x - e>4 Therefore, a plausible relationship between d and e is d = e>4 We now write the actual proof Step 2: Write a proof Let e be given and assume x - d, where d = e>4 The aim is to show that 14x - 152 - e for all x such that x - d We simplify 14x - 152 - and isolate the x - term: 14x - 152 - = 4x - 16 = 0x - 40             (1)1*           less than d = e>4    We have shown that for any e 0, e 4a b = e ƒ1x2 - L = 14x - 152 - e whenever x - d, provided d … e>4 Therefore, lim 14x - 152 = xS4 Related Exercises 19–20 EXAMPLE 5  Limit of a quadratic function  Prove that lim x = 25 using the precise xS5 definition of a limit SOLUTION Step 1: Find d Given e 0, our task is to find an expression for d that depends only on e, such that x - 25 e whenever x - d We begin by factoring x - 25 : x - 25 = 1x + 521x - 52   Factor = x + 0 x -   ab = a 0 b Because the value of d in the inequality x - d typically represents a small positive number, let’s assume d … so that x - 1, which implies that -1 x - or x 6 It follows that x is positive, x + 11, and x - 25 = x + 0 x - 11 x - M02_BRIG3644_03_SE_C02_056-130.indd 121 23/06/17 9:55 AM 122 Chapter 2  •  Limits ➤ The minimum value of a and b is denoted 5a, b6 If x = 5a, b6, then x is the smaller of a and b If a = b, then x equals the common value of a and b In either case, x … a and x … b Using this inequality, we have x - 25 e, provided 11 x - e or x - e>11 Note that two restrictions have been placed on x - : e x - and x - 11 To ensure that both these inequalities are satisfied, let d = 51, e>116 so that d equals the smaller of and e>11 Step 2: Write a proof Let e be given and assume x - d, where d = min51, e>116 By factoring x2 - 25, we have x - 25 = x + 0 x - Because x - d and d … e>11, we have x - e>11 It is also the case that x - because d … 1, which implies that -1 x - or x 6 Therefore, x + 11 and x - 25 = x + 0 x - 11a e b = e 11 We have shown that for any e 0, x2 - 25 e whenever x - d, provided d = min51, e>116 Therefore, lim x2 = 25 xS5 Related Exercises 27, 29 Justifying Limit Laws The precise definition of a limit is used to prove the limit laws in Theorem 2.3 Essential in several of these proofs is the triangle inequality, which states that x + y … x + y , for all real numbers x and y EXAMPLE 6  Proof of Limit Law 1  Prove that if lim ƒ1x2 and lim g1x2 exist, then xSa lim 1ƒ1x2 + g1x22 = lim ƒ1x2 + lim g1x2 xSa ➤ Because lim ƒ1x2 exists, if there exists xSa a d for any given e 0, then there also exists a d for any given 2e xSa xSa xSa SOLUTION  Assume e is given Let lim ƒ1x2 = L, which implies there exists a d1 such that xSa e ƒ1x2 - L whenever x - a d1 e whenever x - a d2 Similarly, let lim g1x2 = M, which implies there exists a d2 such that xSa g1x2 - M Let d = 5d1, d2 and suppose x - a d Because d … d1, it follows that x - a d1 and ƒ1x2 - L e>2 Similarly, because d … d2, it follows that x - a d2 and g1x2 - M e>2 Therefore, 1ƒ1x2 + g1x22 - 1L + M2 = 1ƒ1x2 - L2 + 1g1x2 - M2   Rearrange terms … ƒ1x2 - L + g1x2 - M e e + = e 2 ➤ Proofs of other limit laws are outlined in Exercises 43–44 ➤ Notice that for infinite limits, N plays the role that e plays for regular limits It sets a tolerance or bound for the function values ƒ1x2 M02_BRIG3644_03_SE_C02_056-130.indd 122   Triangle inequality We have shown that given any e 0, if x - a d, then 1ƒ1x2 + g1x22 - 1L + M2 e, which implies that lim 1ƒ1x2 + g1x22 = xSa L + M = lim ƒ1x2 + lim g1x2 xSa xSa Related Exercises 43–44 Infinite Limits In Section 2.4, we stated that lim ƒ1x2 = ∞ if ƒ1x2 grows arbitrarily large as x approaches xSa a More precisely, this means that for any positive number N (no matter how large), ƒ1x2 is larger than N if x is sufficiently close to a but not equal to a 17/07/17 11:16 AM 2.7  Precise Definitions of Limits 123 DEFINITION  Two-Sided Infinite Limit The infinite limit lim ƒ1x2 = ∞ means that for any positive number N, there xSa exists a corresponding d such that ➤ Precise definitions for lim ƒ1x2 = - ∞, ƒ1x2 N whenever x - a d xSa lim+ ƒ1x2 = - ∞, lim+ ƒ1x2 = ∞, xSa xSa lim ƒ1x2 = - ∞, and lim- ƒ1x2 = ∞ x S a- xSa are given in Exercises 57–63 As shown in Figure 2.69, to prove that lim ƒ1x2 = ∞, we let N represent any posixSa tive number Then we find a value of d 0, depending only on N, such that ƒ1x2 N whenever x - a d This process is similar to the two-step process for finite limits y f (x) f (x) N N O a2d x a a1d x , ux au , d Values of x such that f (x) N Figure 2.69 Steps for proving that lim ƒ1 x2 = H xSa Find D Let N be an arbitrary positive number Use the statement ƒ1x2 N to find an inequality of the form x - a d, where d depends only on N Write a proof For any N 0, assume x - a d and use the relationship between N and d found in Step to prove that ƒ1x2 N EXAMPLE 7  An Infinite Limit Proof Let ƒ1x2 = Prove that 1x - 22 lim ƒ1x2 = ∞ xS2 SOLUTION N to find d, 1x - 22 where d depends only on N Taking reciprocals of this inequality, it follows that Step 1: Find d Assuming N 0, we use the inequality N 0x - 20   Take the square root of both sides 1N 1x - 22 ➤ Recall that 2x = x M02_BRIG3644_03_SE_C02_056-130.indd 123 1 has the form x - d if we let d = 1N 1N We now write a proof based on this relationship between d and N The inequality x - 23/06/17 9:55 AM 124 Chapter 2  •  Limits QUICK CHECK 3  In Example 7, if N is increased by a factor of 100, how must d change? Step 2: Write a proof Suppose N is given Let d = and assume 1N 0x - 20 d = Squaring both sides of the inequality 1N 0x - 20 and taking reciprocals, we have 1N 1x - 22   Square both sides N N.  Take reciprocals of both sides 1x - 22 We see that for any positive N, if x - d = , then 1N 1 ƒ1x2 = N It follows that lim = ∞ Note that because x S 1x - 22 1x - 22 d = , d decreases as N increases 1N Related Exercises 45–46 Limits at Infinity Precise definitions can also be written for the limits at infinity lim ƒ1x2 = L and xS ∞ lim ƒ1x2 = L For discussion and examples, see Exercises 64–65 x S -∞ SECTION 2.7 EXERCISES Getting Started y Suppose x lies in the interval 11, 32 with x ≠ Find the smallest positive value of d such that the inequality x - d is true Suppose ƒ1x2 lies in the interval 12, 62 What is the smallest value of e such that ƒ1x2 - e? y f (x) Which one of the following intervals is not symmetric about x = 5? a 11, 92  b.  14, 62  c.  13, 82  d.  14.5, 5.52 Suppose a is a constant and d is a positive constant Give a geometric description of the sets 5x: x - a d6 and 5x: x - a d6 State the precise definition of lim ƒ1x2 = L xSa Interpret ƒ1x2 - L e in words Suppose ƒ1x2 - 0.1 whenever x Find all values of d such that ƒ1x2 - 0.1 whenever x - d Give the definition of lim ƒ1x2 = ∞ and interpret it using xSa pictures Determining values of D from a graph The function ƒ in the figure satisfies lim ƒ1x2 = Determine the largest value of d 0 x 10 Determining values of D from a graph The function ƒ in the figure satisfies lim ƒ1x2 = Determine the largest value of d xS2 satisfying each statement a If x - d, then ƒ1x2 - b If x - d, then ƒ1x2 - 1>2 y y f (x) xS2 satisfying each statement a If x - d, then ƒ1x2 - b If x - d, then ƒ1x2 - M02_BRIG3644_03_SE_C02_056-130.indd 124 x 23/06/17 9:55 AM 2.7  Precise Definitions of Limits 125 Practice Exercises a e = 11 Determining values of D from a graph The function ƒ in the figure satisfies lim ƒ1x2 = Determine the largest value of d b.  e = c For any e 0, make a conjecture about the corresponding values of d satisfying (2) xS3 satisfying each statement a If x - d, then ƒ1x2 - 6 y b If x - d, then ƒ1x2 - 6 y y f (x) y f (x) x 16 Finding a symmetric interval The function ƒ in the figure satisfies lim ƒ1x2 = For each value of e, find all values of xS4 d such that 1 ƒ1x2 - e whenever x - d.(3) x b e = a e = c For any e 0, make a conjecture about the corresponding values of d satisfying (3) 12 Determining values of D from a graph The function ƒ in the figure satisfies lim ƒ1x2 = Determine the largest value of d xS4 satisfying each statement y a If x - d, then ƒ1x2 - b If x - d, then ƒ1x2 - 0.5 y y f (x) T T x 2x - and note x - that lim ƒ1x2 = For each value of e, use a graphing utility to 17 Finding a symmetric interval Let ƒ1x2 = xS1 find all values of d such that ͉ ƒ1x2 - ͉ e whenever ͉ x - ͉ d x b e = a e = c For any e 0, make a conjecture about the value of d that satisfies the preceding inequality 13 Finding D for a given e using a graph Let ƒ1x2 = x + and note that lim ƒ1x2 = For each value of e, use a graphing utility xS0 to find all values of d such that ƒ1x2 - e whenever x - 0 d Sketch graphs illustrating your work a e = T b.  e = 0.5 14 Finding D for a given E using a graph Let g1x2 = 2x - 12x + 26x + and note that lim g1x2 = 24 xS2 For each value of e, use a graphing utility to find all values of d such that g1x2 - 24 e whenever x - d Sketch graphs illustrating your work a e = b.  e = 0.5 15 Finding a symmetric interval The function ƒ in the figure satisfies lim ƒ1x2 = For each value of e, find all values of d xS2 such that ƒ1x2 - e whenever x - d.(2) M02_BRIG3644_03_SE_C02_056-130.indd 125 T 18 Finding a symmetric interval Let ƒ1x2 = c 3x + if x … 2x if x + and note that lim ƒ1x2 = For each value of e, use a graphxS3 ing utility to find all values of d such that ͉ ƒ1x2 - ͉ e whenever ͉ x - ͉ d b e = 14 a e = 12 c For any e 0, make a conjecture about the value of d that satisfies the preceding inequality 19–42 Limit proofs Use the precise definition of a limit to prove the following limits Specify a relationship between e and d that guarantees the limit exists 19 lim 18x + 52 = 13 xS1 20 lim 1- 2x + 82 = xS3 23/06/17 9:55 AM 126 Chapter 2  •  Limits 21 lim x - 16 = (Hint: Factor and simplify.) x - 45–48 Limit proofs for infinite limits Use the precise definition of infinite limits to prove the following limits 22 lim x - 7x + 12 = -1 x - 45 lim xS4 xS3 23 lim x = xS0 xS4 24 lim 5x = xS0 25 lim ƒ1x2 = 9, where ƒ1x2 = b xS7 3x - 12 x + 26 lim ƒ1x2 = 4, where ƒ1x2 = b xS5 2x - - 4x + 24 47 lim a xS0 if x … if x 7 if x … if x = ∞ 1x - 42 46 lim xS - 1 + 1b = ∞ x2 48 lim a xS0 = ∞ 1x + 12 - sin xb = ∞ x4 49 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 lim ƒ1x2 = L xSa a For a given e 0, there is one value of d for which ƒ1x2 - L e whenever x - a d 27 lim x = (Hint: Use the identity 2x = x ) b The limit lim ƒ1x2 = L means that given an arbitrary d 0, 29 lim 1x + 3x2 = 10 c The limit lim ƒ1x2 = L means that for any arbitrary e 0, xS0 xSa we can always find an e such that ƒ1x2 - L e whenever x - a d 28 lim 1x - 32 = (Hint: Use the identity 2x = x ) 2 xS3 xS2 30 lim 12x - 4x + 12 = 17 xSa xS4 31 lim 2x = (Hint: Use the inequality } a - b } … a - b , we can always find a d such that ƒ1x2 - L e whenever x - a d xS - d If x - a d, then a - d x a + d which holds for all constants a and b (see Exercise 74).) 32 lim 1x = (Hint: The factorization x - 25 = x S 25 x - 25 11x - 5211x + 52 implies that 1x - =  .) 1x + 1 33 lim = (Hint: As x S 3, eventually the distance between xS3 x x and is less than Start by assuming x - and show 1 ) 0x0 x - = (Hint: Multiply the numerator and denominax S 1x - tor by 1x + 2.) 34 lim 35 = 10 (Hint: To find d, you need to bound x away x 1 `  .) from So let ` x 10 20 lim x S 1>10 36 lim x sin xS0 = x 37 lim 1x + x 42 = (Hint: You may use the fact that if x c, xS0 50 Finding D algebraically Let ƒ1x2 = x - 2x + a For e = 0.25, find the largest value of d satisfying the statement ƒ1x2 - e whenever x - d b Verify that lim ƒ1x2 = as follows For any e 0, find the xS1 largest value of d satisfying the statement ƒ1x2 - e whenever x - d 51–55 Precise definitions for left- and right-sided limits Use the following definitions Assume ƒ exists for all x near a with x a We say that the limit of ƒ1x2 as x approaches a from the right of a is L and write lim+ ƒ1x2 = L, if for any e there exists d such that xSa ƒ1x2 - L e whenever x - a d Assume ƒ exists for all x near a with x a We say that the limit of ƒ1x2 as x approaches a from the left of a is L and write lim- ƒ1x2 = L, if for any e there exists d such that xSa ƒ1x2 - L e whenever a - x d then x c2.) 38 lim b = b, for any constants a and b xSa 39 lim 1mx + b2 = ma + b, for any constants a, b, and m xSa 40 lim x = 27 xS3 42 lim xS5 41 lim x = xS1 1 = 25 x 51 Comparing definitions Why is the last inequality in the definition of lim ƒ1x2 = L, namely, x - a d, replaced with xSa x - a d in the definition of lim+ ƒ1x2 = L? xSa 52 Comparing definitions Why is the last inequality in the definition of lim ƒ1x2 = L, namely, ͉ x - a ͉ d, replaced with xSa a - x d in the definition of lim- ƒ1x2 = L? xSa 43 Proof of Limit Law Suppose lim ƒ1x2 = L and lim g1x2 = M xSa Prove that lim 1ƒ1x2 - g1x22 = L - M xSa 53 One-sided limit proofs Prove the following limits for xSa ƒ1x2 = b 44 Proof of Limit Law Suppose lim ƒ1x2 = L Prove that xSa lim 1cƒ1x22 = cL, where c is a constant xSa M02_BRIG3644_03_SE_C02_056-130.indd 126 3x - 2x - if x if x Ú a lim+ ƒ1x2 = -4  b.  lim- ƒ1x2 = - 4  c.  lim ƒ1x2 = - xS0 xS0 xS0 27/06/17 11:11 AM 2.7  Precise Definitions of Limits 127 54 Determining values of D from a graph The function ƒ in the figure satisfies lim+ ƒ1x2 = and lim- ƒ1x2 = Determine all xS2 xS2 values of d that satisfy each statement a ƒ1x2 - 0 2 whenever x - d 64–65 Definition of a limit at infinity The limit at infinity lim ƒ1x2 = L means that for any e 0, there exists N such that xS ∞ ƒ1x2 - L e whenever x N Use this definition to prove the following statements b ƒ1x2 - 0 1 whenever x - d c ƒ1x2 - 2 whenever - x d 64 lim xS ∞ d ƒ1x2 - 1 whenever - x d N such that 2x + = x ƒ1x2 M whenever x N Use this definition to prove the following statements y f (x) 66 lim xS ∞ x 55 One-sided limit proof Prove that lim+ 1x = xS0 Explorations and Challenges 56 The relationship between one-sided and two-sided limits Prove the following statements to establish the fact that lim ƒ1x2 = L if xSa and only if lim- ƒ1x2 = L and lim+ ƒ1x2 = L xSa xS ∞ xS ∞ 65 lim 66–67 Definition of infinite limits at infinity We write lim ƒ1x2 = ∞ if for any positive number M, there is a corresponding y 10 = x xSa x = ∞ 100 67 lim xS ∞ x2 + x = ∞ x 68 Proof of the Squeeze Theorem Assume the functions ƒ, g, and h satisfy the inequality ƒ1x2 … g1x2 … h1x2, for all x near a, except possibly at a Prove that if lim ƒ1x2 = lim h1x2 = L, xSa xSa then lim g1x2 = L xSa 69 Limit proof Suppose ƒ is defined for all x near a, except possibly at a Assume for any integer N 0, there is another integer M such that ƒ1x2 - L 1>N whenever x - a 1>M Prove that lim ƒ1x2 = L using the precise definition of a limit xSa 70–72 Proving that lim ƒ1 x2 L Use the following definition for xSa a If lim- ƒ1x2 = L and lim+ ƒ1x2 = L, then lim ƒ1x2 = L the nonexistence of a limit Assume ƒ is defined for all x near a, except b If lim ƒ1x2 = L, then lim- ƒ1x2 = L and lim+ ƒ1x2 = L possibly at a We write lim ƒ1x2 ≠ L if for some e 0, there is no xSa xSa xSa xSa xSa xSa 57 Definition of one-sided infinite limits We write lim+ ƒ1x2 = - ∞ xSa value of d satisfying the condition xSa ƒ1x2 - L e whenever x - a d if for any negative number N, there exists d such that ƒ1x2 N whenever x - a d a Write an analogous formal definition for lim+ ƒ1x2 = ∞ xSa 70 For the following function, note that lim ƒ1x2 ≠ Find all xS2 values of e for which the preceding condition for nonexistence is satisfied b Write an analogous formal definition for lim- ƒ1x2 = - ∞ y c Write an analogous formal definition for lim- ƒ1x2 = ∞ xSa xSa 58–59 One-sided infinite limits Use the definitions given in Exercise 57 to prove the following infinite limits 58 lim+ xS1 = - ∞ - x 59 limxS1 = ∞ - x y f (x) 60–61 Definition of an infinite limit We write lim ƒ1x2 = - ∞ if for any negative number M, there exists d such that xSa ƒ1x2 M whenever x - a d x Use this definition to prove the following statements 60 lim xS1 -2 = - ∞ 1x - 12 61 lim xS - -10 = -∞ 1x + 22 62 Suppose lim ƒ1x2 = ∞ Prove that lim 1ƒ1x2 + c2 = ∞ for any xSa xSa constant c 63 Suppose lim ƒ1x2 = ∞ and lim g1x2 = ∞ Prove that xSa lim 1ƒ1x2 + g1x22 = ∞ xSa M02_BRIG3644_03_SE_C02_056-130.indd 127 xSa 71 Prove that lim xS0 72 Let 0x0 x does not exist ƒ1x2 = b if x is rational if x is irrational Prove that lim ƒ1x2 does not exist for any value of a (Hint: Assume xSa lim ƒ1x2 = L for some values of a and L, and let e =  .) xSa 14/04/18 10:33 AM 128 Chapter 2  •  Limits 73 A continuity proof Suppose ƒ is continuous at a and defined for all x near a If ƒ1a2 0, show that there is a positive number d for which ƒ1x2 for all x in 1a - d, a + d2 (In other words, ƒ is positive for all x in some interval containing a.) QUICK CHECK ANSWERS 1.  d … 50   2.  d … 0.62  3.  d must decrease by a factor of 1100 = 10 (at least) 74 Show that }a - b} … a - b for all constants a and b (Hint: Write a = 1a - b2 + b and apply the triangle inequality to 1a - b2 + b ) CHAPTER 2  REVIEW EXERCISES Explain why or why not Determine whether the following statements are true and give an explanation or counterexample a The rational function y x - has vertical asymptotes at x2 - x = - and x = b Numerical or graphical methods always produce good estimates of lim ƒ1x2 xSa c The value of lim ƒ1x2, if it exists, is found by calculating ƒ1a2 xSa d If lim ƒ1x2 = ∞ or lim ƒ1x2 = - ∞ , then lim ƒ1x2 does not xSa xSa exist xSa e If lim ƒ1x2 does not exist, then either lim ƒ1x2 = ∞ or xSa xSa lim ƒ1x2 = - ∞ y f (x) 22 21 x xSa f The line y = 2x + is a slant asymptote of the function 2x + x ƒ1x2 =   x - g If a function is continuous on the intervals 1a, b2 and 3b, c2, where a b c, then the function is also continuous on 1a, c2 Points of discontinuity Use the graph of ƒ in the figure to determine the values of x in the interval 1-3, 52 at which ƒ fails to be continuous Justify your answers using the continuity checklist y h If lim ƒ1x2 can be calculated by direct substitution, then ƒ is xSa T The height above the ground of a stone thrown upwards is given by s1t2, where t is measured in seconds After 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 s112 and s11.52, and then find the average velocity of the stone over the time interval 31, 1.54 23 22 21 T A baseball is thrown upwards into the air; its distance above the ground after t seconds is given by s1t2 = - 16t + 60t + 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 b lim - ƒ1x2 c lim + ƒ1x2 d lim ƒ1x2 e ƒ112 f lim ƒ1x2 i.  lim+ ƒ1x2 j lim ƒ1x2 xS3 xS - xS1 xS3 xS - g lim ƒ1x2 xS2 x Computing a limit graphically and analytically sin 2u with a graphing utility Comment on any insin u accuracies in the graph and then sketch an accurate graph of the function sin 2u b Estimate lim using the graph in part (a) S u sin u c Verify your answer to part (b) by finding the value of sin 2u analytically using the trigonometric identity lim S u sin u sin 2u = sin u cos u a Graph y = Estimating limits graphically Use the graph of ƒ in the figure to evaluate the function or analyze the limit a ƒ1-12 y f (x) continuous at x = a xS - h lim- ƒ1x2 xS3 T Computing a limit numerically and analytically a Estimate lim x S p>4 cos 2x by making a table of values of cos x - sin x cos 2x for values of x approaching p>4 Round your cos x - sin x estimate to four digits cos 2x b Use analytic methods to find the value of lim x S p>4 cos x - sin x M02_BRIG3644_03_SE_C02_056-130.indd 128 17/07/17 11:16 AM 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) x S -∞ 39 lim a xS ∞ c Evaluate lim- ƒ1t2 and lim+ ƒ1t2 tS3 xS - lim ƒ1x2 = x S 3- xS - lim 18p2 43 lim x - + x2 and  lim x - + x2 x S -∞ ƒ132 = 44 lim x S -∞ xS1 14 lim xSa x - a 3 16 lim x - 7x + 12x - x 17 lim - x2 x - 8x + 18 lim 13x + 16 - x - xS1 19 lim xS3 xS4 xS3 23 lim x S 81 2x - x - 81 24 lim u S p>2 sin2 u - sin u + sin2 u - 1 - 1sin x 25 lim x S p>2 x + p>2 26 lim+ xS1 27 lim xS5 29 limxS3 31 lim+ xS1 33 limxS0 x - x -  and  limS Ax - x Ax - x - x1x - 52 x - x - 3x 4x - 4x 0x - 10 tan x 2x - 35 lim x S ∞ 4x + 10 28 xS ∞ cos t e3t 51 limxS1 x ln x cos4 x b x + x + 53 An important limit a Use a graphing utility to illustrate the inequalities cos x … on 3-1, 14 sin x … x cos x b Use part (a) to explain why lim T xS0 sin x = x x - 5x +   x - 2x a Analyze lim- ƒ1x2, lim+ ƒ1x2, lim- ƒ1x2, and lim+ ƒ1x2 54 Finding vertical asymptotes Let ƒ1x2 = xS0 xS0 xS2 xS2 b Does the graph of ƒ have any vertical asymptotes? Explain c Graph ƒ using a graphing utility and then sketch the graph with paper and pencil, correcting any errors obtained with the graphing utility 55–60 End behavior Evaluate lim ƒ1x2 and lim ƒ1x2 xS ∞ lim + xS - 30 lim+ uS0 32 limxS2 34 lim xS ∞ x - x + u - sin u 2x - x - 5x + 4x + 3x + 28x + x4 - 36 lim xS ∞ x + x S -∞ 4x + 1 - x3 56 ƒ1x2 = 57 ƒ1x2 = - e-2x 58 ƒ1x2 = 6ex + 20 3ex + 60 ƒ1x2 = 55 ƒ1x2 = 59 ƒ1x2 = 61–65 Slant asymptotes x6 + 216x 14 + 1 ln x x + 29x + x a Analyze lim ƒ1x2 and lim ƒ1x2 for each function xS ∞ x S -∞ b Determine whether ƒ has a slant asymptote If so, write the equation of the slant asymptote 61 ƒ1x2 = M02_BRIG3644_03_SE_C02_056-130.indd 129 49 lim a5 + xS0 T p5 - 22 lim pS1 p - 1  and  lim r + er rS ∞ + e r S -∞ 52 Assume the function g satisfies the inequality … g1x2 … sin2 x + 1, for all values of x near Find lim g1x2 t - 1>3 1 a - b 20 lim x - 1x + t S 1>3 13t - 12 x - 81 21 lim S x x - 2e4r + 3e5r 2e4r + 3e5r  and  lim 4r 5r S r -∞ 7e4r - 9e5r 7e - 9e tS ∞ x - 7x + 12x - x x 47 lim 50 lim , where a is constant x - + x2 46 lim x S -∞ 15 lim xS1 xS ∞ 48 lim ex sin x h 13x + 12 - 13a + 12 x  and  lim ln w ln w3 + rS ∞ 1h + 62 + 1h + 62 - 42 hS0 x - + x2 45 lim wS ∞ 15x + 5h - 15x 12 lim , where x is constant hS0 h 13 lim xS ∞ 11 lim 15x + x S 1000 xS ∞ x S -∞ 10–51 Calculating limits Determine the following limits 10 b, where a is constant and a ≠ 41 lim 13 tan-1 x + 22 42 lim 1-3x + 52 xS0 lim ƒ1x2 = b z zS ∞ lim ƒ1x2 = ∞ x S 3+ 2x - ax - 2x - x 40 lim ae-2z + Sketching a graph Sketch the graph of a function ƒ with all the following properties lim + ƒ1x2 = - ∞  , where a is a positive constant xS ∞ d Interpret the meaning of the limits in part (c) e For what values of t is ƒ continuous? Explain lim - ƒ1x2 = ∞ 2ax + 38 lim 2x + ax - 2x - b , where a and b are constants t S 2.9 xS3 3x + 37 lim a Graph the function c = ƒ1t2 that gives the cost of renting a snowboard for t days, for … t … b Evaluate lim ƒ1t2 129 Review Exercises 3x + 5x + x + 62 ƒ1x2 = 9x + 12x - 12 23/06/17 9:55 AM 130 Chapter 2  •  Limits 63 ƒ1x2 = x1x + 22 + x - 2x - x 64 ƒ1x2 = x2 + 3x - 4x 65 ƒ1x2 = 4x + x + x2 - x + 66–68 Finding asymptotes Find all the asymptotes of the following functions T T 66 ƒ1x2 = 2x + 2x + 3x - 68 ƒ1x2 = 2x - x - 67 ƒ1x2 = 71–74 Continuity at a point Use the continuity checklist to determine whether the following functions are continuous at the given value of a 71 ƒ1x2 =  ; a = x - x - 16 72 g1x2 = c x - if x ≠ T  ; a = if x … 2x - if x x - 5x + 6x x - -2 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 2x x - 25x if x = 78 g1x2 = cos ex 79 Determining unknown constants Let p b 84 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 T1t2 equal the temperature t hours after midnight and examine the function ƒ1t2 = T1t2 - T1t + 122, for … t … 12.) T 85 Antibiotic dosing The amount of an antibiotic (in milligrams) in the blood t hours after an intravenous line is opened is given by m1t2 = 1001e-0.1t - e-0.3t2 86 lim 15x - 22 = 87 lim xS5 88 lim ƒ1x2 = 5, where ƒ1x2 = b 89 lim 13x - 42 = x - 25 = 10 x - 3x - -4x + 17 if x … if x 90 lim+ 24x - = xS2 xS2 91 lim = ∞ x S 1x - 22 92 Limit proofs a Assume ͉ ƒ1x2 ͉ … L for all x near a and lim g1x2 = Give a xSa formal proof that lim 1ƒ1x2g1x22 = xSa if x if x = if x Determine values of the constants a and b, if possible, for which g is continuous at x = Chapter Guided Projects 83 x = cos x; a0, xS3 76 g1x2 = e2x - 5x - g1x2 = c a ax + bx T 82 x + 7x + = 0; 1-1, 02 xS1 ;a = 75–78 Continuity on intervals Find the intervals on which the following functions are continuous Specify right- or left-continuity at the finite endpoints 75 ƒ1x2 = 2x - 82–83 Intermediate Value Theorem 86–91 Limit proofs Use an appropriate limit definition to prove the following limits  ; a = if x ≠ 81 Sketch the graph of a function that is continuous on (0, 14 and on 11, 22 but is not continuous on 10, 22 a Use the Intermediate Value Theorem to show that the amount of drug is 30 mg at some time in the interval 30, 54 and again at some time in the interval 35, 154 b Estimate the times at which m = 30 mg c Is the amount of drug in the blood ever 50 mg? if x = - 2x + 14 77 h1x2 = b Is h1x2 = 2x - right-continuous at x = 3? Explain T 70 Finding all asymptotes Find all the asymptotes of x2 + x + Plot a graph of ƒ together with its asympƒ1x2 = 0x0 totes (Hint: Consider the cases x = 0, x 0, and x 0.) 74 g1x2 = c a Is h1x2 = 2x - left-continuous at x = 3? Explain tan-1 x 69 Two slant asymptotes Explain why the function x + xex + 10ex ƒ1x2 = has two slant asymptotes, y = x and 21ex + 12 y = x + Plot a graph of ƒ together with its two slant asymptotes 73 h1x2 = b 80 Left- and right-continuity b Find a function ƒ for which lim 1ƒ1x21x - 222 ≠ Why xS2 doesn’t this violate the result stated in part (a)? c The Heaviside function is defined as H1x2 = b if x if x Ú Explain why lim 1xH1x22 = xS0 Applications of the material in this chapter and related topics can be found in the following Guided Projects For additional information, see the Preface • Fixed-point iteration • Local linearity M02_BRIG3644_03_SE_C02_056-130.indd 130 17/07/17 11:16 AM ... Names: Briggs, William L., author | Cochran, Lyle, author | Gillett, Bernard, author | Schulz, Eric P., author Title: Calculus Early transcendentals Description: Third edition / William Briggs,. .. / William Briggs, University of Colorado, Denver, Lyle Cochran, Whitworth University,   Bernard Gillett, University of Colorado, Boulder, Eric Schulz, Walla Walla Community College | New York,... PM Calculus E A R LY T R A N S C E N D E N TA L S Third Edition WILLIAM BRIGGS University of Colorado, Denver LYLE COCHRAN Whitworth University BERNARD GILLETT University of Colorado, Boulder ERIC