MATH 201 Problems Solution Chapter 2, Fall 2008

CHAPTER 2 LIMITS AND CONTINUITY 2.1 RATES OF CHANGE AND LIMITS 1. (a) Does not exist. As x approaches 1 from the right, g(x) approaches 0. As x approaches 1 from the left, g(x) approaches 1. There is no single number L that all the values g(x) get arbitrarily close to as x Ä 1. (b) 1 (c) 0 2. (a) 0 (b) �1 (c) Does not exist. As t approaches 0 from the left, f(t) approaches 1. As t approaches 0 from the right, f(t) � approaches 1. There is no single number L that f(t) gets arbitrarily close to as t 0. Ä 3. (a) True (b) True (c) False (d) False (e) False (f) True 4. (a) False (b) False (c) True (d) True (e) True 5. lim does not exist because 1 if x 0 and 1 if x 0. As x approaches 0 from the left, x Ä 0 x x x x x x x x x x k k k k k k œ œ � œ œ � � � approaches 1. As x approaches 0 from the right, approaches 1. There is no single number L that all x x x x k k k k � the function values get arbitrarily close to as x 0. Ä 6. As x approaches 1 from the left, the values of become increasingly large and negative. As x approaches 1 " � x 1 from the right, the values become increasingly large and positive. There is no one number L that all the function values get arbitrarily close to as x 1, so lim does not exist. Ä x Ä 1 " � x 1 7. Nothing can be said about f(x) because the existence of a limit as x Ä x does not depend on how the function ! is defined at x . In order for a limit to exist, f(x) must be arbitrarily close to a single real number L when ! x is close enough to x . That is, the existence of a limit depends on the values of f(x) for x x , not on the ! ! near definition of f(x) at x itself. ! 8. Nothing can be said. In order for lim f(x) to exist, f(x) must close to a single value for x near 0 regardless of x Ä 0 the value f(0) itself. 9. No, the definition does not require that f be defined at x œ 1 in order for a limiting value to exist there. If f(1) is defined, it can be any real number, so we can conclude nothing about f(1) from lim f(x) 5. x Ä 1 œ 10. No, because the existence of a limit depends on the values of f(x) when x is near 1, not on f(1) itself. If lim f(x) exists, its value may be some number other than f(1) 5. We can conclude nothing about lim f(x), x 1 x 1 Ä Ä œ whether it exists or what its value is if it does exist, from knowing the value of f(1) alone. Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley 68 Chapter 2 Limits and Continuity 11. (a) f(x) x /(x 3) œ � * � a b # x 3.1 3.01 3.001 3.0001 3.00001 3.000001 f(x) 6.1 6.01 6.001 6.0001 6.00001 6.000001 � � � � � � � � � � � � x 2.9 2.99 2.999 2.9999 2.99999 2.999999 f(x) 5.9 5.99 5.999 5.9999 5.99999 5.999999 � � � � � � � � � � � � The estimate is lim f(x) 6. x Ä �$ œ � (b) (c) f(x) x 3 if x 3, and lim (x 3) 3 3 6. œ œ œ � Á � � œ � � œ � x 9 x 3 x 3 (x 3)(x 3) # � � � � � x Ä �$ 12. (a) g(x) x / x 2 œ � # � a b Š È ‹ # x 1.4 1.41 1.414 1.4142 1.41421 1.414213 g(x) 2.81421 2.82421 2.82821 2.828413 2.828423 2.828426 (b) (c) g(x) x 2 if x 2, and lim x 2 2 2 2 2. œ œ œ � Á � œ � œ x 2 x 2 x 2 x 2 x 2 # � � � � � È Š ‹ Š ‹ È È Š È ‹ È È È È È È Š ‹ x Ä # È 13. (a) G(x) (x 6)/ x 4x 12 œ � � � a b # x 5.9 5.99 5.999 5.9999 5.99999 5.999999 G(x) .126582 .1251564 .1250156 .1250015 .1250001 .1250000 � � � � � � � � � � � � x 6.1 6.01 6.001 6.0001 6.00001 6.000001 G(x) .123456 .124843 .124984 .124998 .124999 .124999 � � � � � � � � � � � � Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley Section 2.1 Rates of Change and Limits 69 (b) (c) G(x) if x 6, and lim 0.125. œ œ œ Á � œ œ � œ � x 6 x 6 x 4x 12 (x 6)(x 2) x x 2 2 8 � � " " " " � � � � � # � �' � a b # x Ä �' 14. (a) h(x) x 2x 3 / x 4x 3 œ � � � � a b a b # # x 2.9 2.99 2.999 2.9999 2.99999 2.999999 h(x) 2.052631 2.005025 2.000500 2.000050 2.000005 2.0000005 x 3.1 3.01 3.001 3.0001 3.00001 3.000001 h(x) 1.952380 1.995024 1.999500 1.999950 1.999995 1.999999 (b) (c) h(x) if x 3, and lim 2. œ œ œ Á œ œ œ x 2x 3 x 1 x 1 3 1 4 x 4x 3 (x 3)(x 1) x 1 x 1 3 1 (x 3)(x 1) # # � � � � � � � � � � � � # � � x Ä $ 15. (a) f(x) x 1 / x 1 œ � � a b a b k k # x 1.1 1.01 1.001 1.0001 1.00001 1.000001 f(x) 2.1 2.01 2.001 2.0001 2.00001 2.000001 � � � � � � x .9 .99 .999 .9999 .99999 .999999 f(x) 1.9 1.99 1.999 1.9999 1.99999 1.999999 � � � � � � (b) Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley 70 Chapter 2 Limits and Continuity (c) f(x) , and lim (1 x) 1 ( 1) 2. x 1, x 0 and x 1 1 x, x 0 and x 1 œ œ � œ � � œ œ � Á œ � � Á � x x 1 (x 1)(x 1) x 1 (x 1)(x 1) (x 1) # � " � � � � � � � � k k � x 1 Ä � 16. (a) F(x) x 3x 2 / 2 x œ � � � a b a k kb # x 2.1 2.01 2.001 2.0001 2.00001 2.000001 F(x) 1.1 1.01 1.001 1.0001 1.00001 1.000001 � � � � � � � � � � � � x 1.9 1.99 1.999 1.9999 1.99999 1.999999 F(x) .9 .99 .999 .9999 .99999 .999999 � � � � � � � � � � � � (b) (c) F(x) , and lim (x 1) 2 1 , x 0 x 1, x 0 and x 2 œ œ � œ � � œ œ � � Á � x 3x 2 2 x (x 2)(x 1) x (x 2)(x ) 2 x # � � � � � # � � � " � k k � x Ä �# �1. 17. (a) g( ) (sin )/ ) ) ) œ .1 .01 .001 .0001 .00001 .000001 g( ) .998334 .999983 .999999 .999999 .999999 .999999 ) ) .1 .01 .001 .0001 .00001 .000001 g( ) .998334 .999983 .999999 .999999 .999999 .999999 ) ) � � � � � � lim g( ) 1 ) Ä ! ) œ (b) 18. (a) G(t) (1 cos t)/t œ � # t .1 .01 .001 .0001 .00001 .000001 G(t) .499583 .499995 .499999 .5 .5 .5 t .1 .01 .001 .0001 .00001 .000001 G(t) .499583 .499995 .499999 .5 .5 .5 � � � � � � lim G(t) 0.5 t Ä ! œ Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley Section 2.1 Rates of Change and Limits 71 (b) Graph is NOT TO SCALE 19. (a) f(x) œ x "ÎÐ"� Ñ x x .9 .99 .999 .9999 .99999 .999999 f(x) .348678 .366032 .367695 .367861 .367877 .367879 x 1.1 1.01 1.001 1.0001 1.00001 1.000001 f(x) .385543 .369711 .368063 .367897 .367881 .367878 lim f(x) 0.36788 x Ä 1 ¸ (b) Graph is NOT TO SCALE. Also the intersection of the axes is not the origin: the axes intersect at the point (1 2.71820). ß 20. (a) f(x) 3 1 /x œ � a b x x .1 .01 .001 .0001 .00001 .000001 f(x) 1.161231 1.104669 1.099215 1.098672 1.098618 1.098612 x .1 .01 .001 .0001 .00001 .000001 f(x) 1.040415 1.092599 1.098009 1.098551 1.098606 1.098611 � � � � � � lim f(x) 1.0986 x Ä ! ¸ (b) 21. lim 2x 2(2) 4 22. lim 2x 2(0) 0 x x Ä # Ä ! œ œ œ œ 23. lim (3x 1) 3 1 0 24. lim x x 1 Ä Ä " $ � œ � œ œ œ � ˆ ‰ " � �" " � � # 3 3x 1 3(1) 1 1 Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley 72 Chapter 2 Limits and Continuity 25. lim 3x(2x 1) 3( 1)(2( 1) 1) 9 26. lim 1 x x 1 Ä � Ä � " � œ � � � œ œ œ œ � 3x 3 2x 1 2( 1) 1 3 3( 1) # # � � � � � 27. lim x sin x sin 28. lim x x Ä Ä 1 # œ œ œ œ œ 1 1 1 1 1 1 1 1 # # # � � � � �" " 1 cos x cos 1 1 1 1 29. (a) 19 (b) 1 ? ? ? ? f 28 9 f 2 0 x 3 1 x 1 ( 1) f(3) f(2) f(1) f( ) œ œ œ œ œ œ � � �" � # � � # � � 30. (a) 0 (b) 2 ? ? ? ? g g(1) g( 1) g g(0) g( 2) x 1 ( 1) 2 x 0 ( 2) 1 1 0 4 œ œ œ œ œ œ � � � � � � � � � # � � 31. (a) (b) ? ? ? 1 ? 1 h 1 1 4 h t t h h h h 0 3 3 3 œ œ œ � œ œ œ ˆ ‰ ˆ ‰ È È ˆ ‰ ˆ ‰ 3 4 4 3 4 4 6 6 3 1 1 1 1 1 1 1 1 1 1 � � � � � � � � # # # 32. (a) (b) 0 ? 1 ? 1 1 ? 1 1 1 ? 1 1 1 g g( ) g(0) (2 1) (2 1) g g( ) g( ) (2 1) (2 ) t 0 0 t ( ) 2 œ œ œ � œ œ œ � � � � � � � � � " � � � � # 33. 1 ? ?) R 3 R(2) R(0) 2 0 8 1 1 œ œ œ œ � � # # � � � " È È 34. 2 2 0 ? ?) P P(2) P(1) (8 16 10) ( ) 2 1 1 œ œ œ � œ � � � � " � % � & � 35. (a) Q Slope of PQ œ ? ? p t Q (10 225) 42.5 m/sec " � � ß œ 650 225 20 10 Q (14 375) 45.83 m/sec # � � ß œ 650 375 20 14 Q (16.5 475) 50.00 m/sec $ � � ß œ 650 475 20 16.5 Q (18 550) 50.00 m/sec % � � ß œ 650 550 20 18 (b) At t 20, the Cobra was traveling approximately 50 m/sec or 180 km/h. œ 36. (a) Q Slope of PQ œ ? ? p t Q (5 20) 12 m/sec " � � ß œ 80 20 10 5 Q (7 39) 13.7 m/sec # � � ß œ 80 39 10 7 Q (8.5 58) 14.7 m/sec $ � � ß œ 80 58 10 8.5 Q (9.5 72) 16 m/sec % � � ß œ 80 72 10 9.5 (b) Approximately 16 m/sec 37. (a) (b) 56 thousand dollars per year ? ? p t 1994 1992 174 62 112 œ œ œ � � # (c) The average rate of change from 1991 to 1992 is 35 thousand dollars per year. ? ? p t 1992 1991 62 27 œ œ � � The average rate of change from 1992 to 1993 is 49 thousand dollars per year. ? ? p t 1993 1992 111 62 œ œ � � So, the rate at which profits were changing in 1992 is approximatley 35 49 42 thousand dollars per year. " #a b � œ Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley Section 2.1 Rates of Change and Limits 73 38. (a) F(x) (x 2)/(x 2) œ � � x 1.2 1.1 1.01 1.001 1.0001 1 F(x) 4.0 3.4 3.04 3.004 3.0004 3 � � � � � � 5.0; 4.4; ? ? ? ? F F x 1.2 1 x 1.1 1 4.0 ( 3) 3.4 ( 3) œ œ � œ œ � � � � � � � � � 4.04; 4. ; ? ? ? ? F F x 1.01 1 x 1.001 1 3.04 ( 3) 3.004 ( 3) œ œ � œ œ � !!% � � � � � � � � 4. ; ? ? F x 1.0001 1 3. ( 3) œ œ � !!!% � !!!% � � � (b) The rate of change of F(x) at x œ 1 is 4. � 39. (a) 0.414213 0.449489 ? ? ? ? g g(2) g(1) 2 g g(1.5) g(1) x 2 1 1 x 1.5 1 0.5 1.5 œ œ ¸ œ œ ¸ � � " � � # � � � " È È ? ? g g(1 h) g(1) x (1 h) 1 h 1 h œ œ � � � � � �" È (b) g(x) x œ È 1 h 1.1 1.01 1.001 1.0001 1.00001 1.000001 1 h 1.04880 1.004987 1.0004998 1.0000499 1.000005 1.0000005 1 h 1 /h 0.4880 0.4987 0.4998 0.499 0.5 � � � � È Š ‹ È 0.5 (c) The rate of change of g(x) at x 1 is 0.5. œ (d) The calculator gives lim . h Ä ! œ È1 h h � �" " # 40. (a) i) f(3) f(2) 3 2 1 1 6 � � � " œ œ œ � " " �" # 3 6 ii) , T 2 f(T) f(2) T T T T(T 2) T(2 T) T 2 T 2 T � � # � # � # # � �# � # � � � � " œ œ œ œ œ � Á " " # # # T T T 2 T (b) T 2.1 2.01 2.001 2.0001 2.00001 2.000001 f(T) 0.476190 0.497512 0.499750 0.4999750 0.499997 0.499999 f(T) f(2) / T 2 0.2381 0.2488 0.2 a b a b � � � � � 500 0.2500 0.2500 0.2500 � � � (c) The table indicates the rate of change is 0.25 at t 2. � œ (d) lim T Ä # ˆ ‰ " " �#T 4 œ � 41-46. Example CAS commands: : Maple f := x -> (x^4 16)/(x 2); � � x0 := 2; plot( f(x), x x0-1..x0+1, color black, œ œ title œ "Section 2.1, #41(a)" ); limit( f(x), x œ x0 ); In Exercise 43, note that the standard cube root, x^(1/3), is not defined for x<0 in many CASs. This can be overcome in Maple by entering the function as f := x -> (surd(x+1, 3) � 1)/x. : (assigned function and values for x0 and h may vary) Mathematica Clear[f, x] f[x_]:=(x x 5x 3)/(x 1) 3 2 2 � � � � x0= 1; h = 0.1; � Plot[f[x],{x, x0 h, x0 h}] � � Limit[f[x], x x0] Ä Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley 74 Chapter 2 Limits and Continuity 2.2 CALCULATING LIMITS USING THE LIMIT LAWS 1. lim (2x 5) 2( 7) 5 14 5 9 2. lim (10 3x) 10 3(12) 10 36 26 x x 1 Ä �( Ä # � œ � � œ � � œ � � œ � œ � œ � 3. lim x 5x 2 (2) 5(2) 2 4 10 2 4 x Ä # a b � � � œ � � � œ � � � œ # # 4. lim x 2x 4x 8 ( 2) 2( 2) 4( 2) 8 8 8 8 8 16 x Ä �# a b $ # $ # � � � œ � � � � � � œ � � � � œ � 5. lim 8(t 5)(t 7) 8(6 5)(6 7) 8 6. lim 3s(2s 1) 3 2 1 2 1 t s Ä ' Ä � � œ � � œ � � œ � œ � œ # $ ˆ ‰ � ‘ ˆ ‰ ˆ ‰ 2 2 4 2 3 3 3 3 7. lim 8. lim 2 x x Ä # Ä & x 3 2 3 5 4 4 4 x 6 2 6 8 x 7 5 7 � � � � � � �# œ œ œ œ œ � 9. lim y Ä �& y ( 5) 5 y 5 ( 5) 10 25 5 # # � � � # � œ œ œ 10. lim y Ä # y 2 y 5y 6 (2) 5( ) 6 4 10 6 0 5 2 2 4 4 � � � � # � � � # � " # # œ œ œ œ 11. lim 3(2x 1) 3(2( 1) 1) 3( 3) 27 x Ä �" � œ � � œ � œ # # # 12. lim (x 3) ( 4 3) ( 1) 1 x Ä �% � œ � � œ � œ "*)% "*)% "*)% 13. lim (5 y) [5 ( 3)] (8) (8) 2 16 y Ä �$ � œ � � œ œ œ œ %Î$ %Î$ %Î$ "Î$ % % ˆ ‰ 14. lim (2z 8) (2(0) 8) ( 8) 2 z Ä ! � œ � œ � œ � "Î$ "Î$ "Î$ 15. lim h Ä ! 3 3 3 3 3h 1 1 3(0) 1 1 1 1 2 È È È � � � � � œ œ œ 16. lim h Ä ! 5 5 5 5 5h 4 2 5(0) 4 2 4 4 È È È � � � � � # œ œ œ 17. lim lim lim lim lim h 0 h 0 h 0 h 0 h 0 Ä Ä Ä Ä Ä È È È È È a b Š ‹ Š ‹ È È 3h 1 3h 1 3h 1 1 3h 1 h h 3h 1 1 3h 1 h 3h 1 h 3h 1 3h 3 � � " � � " � � � " � � � � �" � � " � � " œ † œ œ œ œ œ 3 È" � " $ # 18. lim lim lim lim lim h 0 h 0 h 0 h 0 h 0 Ä Ä Ä Ä Ä È È È È È a b Š ‹ Š ‹ È È 5h 4 2 5h 4 2 5h 4 2 5h 4 4 h h 5h 4 2 5h 4 2 h 5h 4 2 h 5h 4 2 5h 5 � � � � � � � � � � � � � � � � œ † œ œ œ œ œ 5 5 4 2 4 È � 19. lim lim lim x x x Ä & Ä & Ä & x 5 x 5 1 x 25 (x 5)(x 5) x 5 5 5 10 � � " " � � � � � # œ œ œ œ 20. lim lim lim x x x Ä �$ Ä �$ Ä �$ x 3 x 3 1 x 4x 3 (x 3)(x 1) x 1 3 1 2 � � " " � � � � � � � # œ œ œ œ � 21. lim lim lim (x 2) 7 x x x Ä �& Ä �& Ä �& x 3x 0 x 5 x 5 (x 5)(x 2) # � � " � � � � œ œ � œ �& � # œ � Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley Section 2.2 Calculating Limits Using the Limit Laws 75 22. lim lim lim (x 5) 2 5 3 x x x Ä # Ä # Ä # x 7x 0 x x 2 (x 5)(x 2) # � � " � # � � � œ œ � œ � œ � 23. lim lim lim t t t Ä " Ä " Ä " t t 2 t 2 1 2 3 t 1 (t 1)(t 1) t 1 1 1 (t 2)(t 1) # #� � � � � � � � � # � � œ œ œ œ 24. lim lim lim t t t Ä �" Ä �" Ä �" t 3t 2 t 2 1 2 1 t t 2 (t 2)(t 1) t 2 1 2 3 (t 2)(t 1) # #� � � � � � � � � � � � � � œ œ œ œ � 25. lim lim lim x x x Ä �# Ä �# Ä �# � � � � � � � � 2x 4 2 2 1 x 2x x (x 2) x 4 2 2(x 2) $ # # # œ œ œ œ � 26. lim lim lim y 0 y y Ä Ä ! Ä ! 5y 8y y (5y 8) 5y 8 3y 16y y 3y 16 3y 16 16 8 $ # # % # # # # � � � � � � � # " œ œ œ œ � a b 27. lim lim lim u 1 u 1 u 1 Ä Ä Ä u 4 u 1 u u 1 (u 1) u u 1 1 1 1 3 u (u 1)(u 1) u (u 1) (1 1)(1 1) % $ # # # # � " � � � � � � � � � " � � � " � � � œ œ œ œ a b a b a b 28. lim lim lim v v v Ä # Ä # Ä # v 8 v 2v 4 4 4 4 12 3 v 16 (v 2) v 2v 4 (v 2)(v 2) v 4 (v 2) v 4 (4)(8) 32 8 $ # % # # # � � � � � � � � � � � � � � œ œ œ œ œ a b a b a b 29. lim lim lim x x x Ä * Ä * Ä * È È ˆ ‰ ˆ ‰ È È È È x 3 x 3 x 9 6 x 3 x 3 x 3 9 3 � � � � � � " " " � œ œ œ œ 30. lim lim lim lim x 2 x 4(2 2) 16 x x x x Ä % Ä % Ä % Ä % 4x x 2 x 2 x 2 x x(4 x) x 2 x 2 x � � � � � � � # È È È ˆ ‰ ˆ ‰ È È œ œ œ � œ � œ ˆ ‰ È 31. lim lim lim lim x 3 x 1 x 1 x 1 x 1 Ä Ä Ä Ä x 1 x 3 2 x 3 x 3 (x 1) x 3 2 (x 1) x 3 (x 3) 4 � � � � � # � � # � � � � � � # � � È È È ˆ ‰ ˆ ‰ È È ˆ ‰ ˆ ‰ œ œ œ � � # Š ‹ È 4 2 4 œ � œ È 32. lim lim lim x x x 1 Ä � Ä � Ä � " " œ œ È Š ‹ Š ‹ È È Š ‹ Š ‹ È È a b x 8 3 x 1 x 8 x 8 (x 1) x 8 (x 1) x 8 x 8 # # # # # # � � � � � $ � � $ � � � $ � � � $ � � * lim lim œ œ œ œ � x 1 x 1 Ä � Ä � (x 1)(x 1) (x 1) x x 1 2 x 3 3 3 � � � � ) � $ � � " � ) � $ � Š ‹ È È # # 33. lim lim lim x 2 x 2 x 2 Ä Ä Ä È Š ‹ Š ‹ È È Š ‹ Š ‹ È È a b x 12 4 x 2 x 12 4 x 12 4 (x 2) x 12 4 (x 2) x 12 4 x 12 16 # # # # # # � � � � � � � � � � � � � � � œ œ lim lim œ œ œ œ x 2 x 2 Ä Ä (x 2)(x 2) (x 2) x 12 4 x 2 4 x 12 4 16 4 2 � � � � � � " � � � Š ‹ È È È # # 34. lim lim lim x 2 x 2 x 2 Ä � Ä � Ä � x 2 x 5 3 x 2 x 5 3 x 2 x 5 3 x 5 3 x 5 3 x 5 9 � � � � � � � � � � � � � � � È a b a b Š ‹ Š ‹ È È Š ‹ Š ‹ È È a b # # # # # # œ œ lim lim œ œ œ œ � x 2 x 2 Ä � Ä � a b Š ‹ È È È x 2 x 5 3 (x 2)(x 2) x 2 4 2 x 5 3 9 3 3 � � � � � � � � � � # # 35. lim lim lim x 3 x 3 x 3 Ä � Ä � Ä � 2 x 5 x 3 2 x 5 2 x 5 (x 3) 2 x 5 (x 3) 2 x 5 4 x 5 � � � � � � � � � � � � � � � È Š ‹ Š ‹ È È Š ‹ Š ‹ È È a b # # # # # # œ œ lim lim lim œ œ œ œ œ x 3 x 3 x 3 Ä � Ä � Ä � 9 x 3 x 6 3 (x 3) 2 x 5 (x 3) 2 x 5 (3 x)(3 x) 2 x 5 2 4 2 � � � � � � � � � � � � � # # # # Š ‹ Š ‹ È È È È Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley 76 Chapter 2 Limits and Continuity 36. lim lim lim x 4 x 4 x 4 Ä Ä Ä 4 x 5 x 9 4 x 5 x 9 4 x 5 x 9 5 x 9 5 x 9 25 x 9 � � � � � � � � � � � � � � � È a b a b Š ‹ Š ‹ È È Š ‹ Š ‹ È È a b # # # # # # œ œ lim lim lim œ œ œ œ œ x 4 x 4 x 4 Ä Ä Ä a b a b Š ‹ Š ‹ È È È È 4 x 5 x 9 4 x 5 x 9 16 x (4 x)(4 x) 4 x 8 4 5 x 9 5 25 5 � � � � � � � � � � � � � # # # # 37. (a) quotient rule (b) difference and power rules (c) sum and constant multiple rules 38. (a) quotient rule (b) power and product rules (c) difference and constant multiple rules 39. (a) lim f(x) g(x) lim f(x) lim g(x) (5)( 2) 10 x c x c x c Ä Ä Ä œ œ � œ � ’ “ ’ “ (b) lim 2f(x) g(x) 2 lim f(x) lim g(x) 2(5)( 2) 20 x c x c x c Ä Ä Ä œ œ � œ � ’ “ ’ “ (c) lim [f(x) 3g(x)] lim f(x) 3 lim g(x) 5 3( 2) 1 x c x c x c Ä Ä Ä � œ � œ � � œ � (d) lim x Ä c f(x) f(x) g(x) lim f(x) lim g(x) 5 ( 2) 7 lim f(x) 5 5 � � � � œ œ œ x c x c x c Ä Ä Ä 40. (a) lim [g(x) 3] lim g(x) lim 3 x x x Ä % Ä % Ä % � œ � œ �$ � $ œ ! (b) lim xf(x) lim x lim f(x) (4)(0) 0 x x x Ä % Ä % Ä % œ œ œ † (c) lim [g(x)] lim g(x) [ 3] 9 x x Ä % Ä % # # # œ œ � œ ’ “ (d) lim 3 x Ä % g(x) f(x) 1 lim f(x) lim 1 0 1 lim g(x) 3 � � � � œ œ œ x x x Ä% Ä% Ä% 41. (a) lim [f(x) g(x)] lim f(x) lim g(x) 7 ( 3) 4 x b x b x b Ä Ä Ä � œ � œ � � œ (b) lim f(x) g(x) lim f(x) lim g(x) (7)( 3) 21 x b x b x b Ä Ä Ä † œ œ � œ � ’ “ ’ “ (c) lim 4g(x) lim 4 lim g(x) (4)( 3) 12 x b x b x b Ä Ä Ä œ œ � œ � ’ “ ’ “ (d) lim f(x)/g(x) lim f(x)/ lim g(x) x b x b x b Ä Ä Ä œ œ œ � 7 7 3 3 � 42. (a) lim [p(x) r(x) s(x)] lim p(x) lim r(x) lim s(x) 4 0 ( 3) 1 x x x x Ä �# Ä �# Ä �# Ä �# � � œ � � œ � � � œ (b) lim p(x) r(x) s(x) lim p(x) lim r(x) lim s(x) (4)(0)( 3) 0 x x x x Ä �# Ä �# Ä �# Ä �# † † œ œ � œ ’ “ ’ “ ’ “ (c) lim [ 4p(x) 5r(x)]/s(x) 4 lim p(x) 5 lim r(x) lim s(x) [ 4(4) 5(0)]/ 3 x x x x Ä �# Ä �# Ä �# Ä �# � � œ � � œ � � � œ ’ “ ‚ "6 3 43. lim lim lim lim (2 h) 2 h h h h Ä ! Ä ! Ä ! Ä ! (1 h) 1 h(2 h) h h h 1 2h h 1 � � � � � � # # # œ œ œ � œ 44. lim lim lim lim (h 4) 4 h h h h Ä ! Ä ! Ä ! Ä ! ( 2 h) ( 2) h(h 4) h h h 4 4h h 4 � � � � � � � � # # # œ œ œ � œ � 45. lim lim 3 h h Ä ! Ä ! [3(2 h) 4] [3(2) 4] h h 3h � � � � œ œ 46. lim lim lim lim h h h h Ä ! Ä ! Ä ! Ä ! ˆ ‰ ˆ ‰ " " � �# � �# � � h 2 2 h � �" � � �# � � � � � � � " h 2h 2h( h) h(4 2h) 4 2 ( 2 h) h œ œ œ œ � Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley Section 2.2 Calculating Limits Using the Limit Laws 77 47. lim lim lim h h h Ä ! Ä ! Ä ! È È Š ‹ Š ‹ È È È È Š ‹ Š ‹ È È È È 7 h 7 h 7 h 7 7 h 7 h 7 h 7 h 7 h 7 (7 h) 7 � � � � � � � � � � � � œ œ lim lim œ œ œ h h Ä ! Ä ! h h 7 h 7 7 h 7 7 Š ‹ È È È È È � � " " � � # 48. lim lim lim h h h Ä ! Ä ! Ä ! È È Š ‹ Š ‹ È È Š ‹ Š ‹ È È 3(0 h) 1 3(0) 1 h 3h 1 3h 1 h 3h 1 h 3h 1 1 (3h 1) � � � � � � " � � " � � " � � � � " œ œ lim lim œ œ œ h h Ä ! Ä ! 3h 3 3 h 3h 1 3h 1 1 Š ‹ È È � � " � � # 49. lim 5 2x 5 2(0) 5 and lim 5 x 5 (0) 5; by the sandwich theorem, x x Ä ! Ä ! È È È È È È � œ � œ � œ � œ # # # # lim f(x) 5 x Ä ! œ È 50. lim 2 x 2 0 2 and lim 2 cos x 2(1) 2; by the sandwich theorem, lim g(x) 2 x x x Ä ! Ä ! Ä ! a b � œ � œ œ œ œ # 51. (a) lim 1 1 1 and lim 1 1; by the sandwich theorem, lim 1 x x x Ä ! Ä ! Ä ! Š ‹ � œ � œ œ œ x 0 x sin x 6 6 2 2 cos x # � (b) For x 0, y (x sin x)/(2 2 cos x) Á œ � lies between the other two graphs in the figure, and the graphs converge as x 0. Ä 52. (a) lim lim lim 0 and lim ; by the sandwich theorem, x x x x Ä ! Ä ! Ä ! Ä ! Š ‹ " " " " " # # # # # # # � œ � œ � œ œ x 1 x 24 4 # # lim . x Ä ! 1 cos x x � " # # œ (b) For all x 0, the graph of f(x) (1 cos x)/x Á œ � # lies between the line y œ " and the parabola # y x /24, and the graphs converge as x 0. œ � Ä " # # 53. lim f(x) exists at those points c where lim x lim x . Thus, c c c 1 c 0 x c x c x c Ä Ä Ä % # % # # # œ œ Ê � œ a b c 0, 1, or 1. Moreover, lim f(x) lim x 0 and lim f(x) lim f(x) 1. Ê œ � œ œ œ œ x x x 1 x 1 Ä ! Ä ! Ä � Ä # 54. Nothing can be concluded about the values of f, g, and h at x œ 2. Yes, f(2) could be 0. Since the conditions of the sandwich theorem are satisfied, lim f(x) 5 0. x Ä # œ � Á 55. 1 lim lim f(x) 5 2(1) lim f(x) 2 5 7. œ œ œ Ê � œ Ê œ � œ x x x Ä % Ä % Ä % f(x) 5 x 2 lim x lim 2 lim f(x) lim 5 lim f(x) 5 � � � % � # � � x x x x x Ä% Ä% Ä% Ä% Ä% Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley 78 Chapter 2 Limits and Continuity 56. (a) 1 lim lim f(x) 4. œ œ œ Ê œ x x Ä �# Ä �# f(x) x lim x lim f(x) lim f(x) # # x x x Ä�# Ä�# Ä�# % (b) 1 lim lim lim lim lim 2. œ œ œ Ê œ � x x x x x Ä �# Ä �# Ä �# Ä �# Ä �# f(x) f(x) f(x) f(x) x x x x x # ’ “ ’ “ ’ “ ˆ ‰ " " �# 57. (a) 0 3 0 lim lim (x 2) lim (x 2) lim [f(x) 5] lim f(x) 5 œ œ � œ � œ � œ � † ’ “ ’ “ ’ “ Š ‹ x x x x x Ä # Ä # Ä # Ä # Ä # f(x) 5 f(x) 5 x x � � � # � # lim f(x) 5. Ê œ x Ä # (b) 0 4 0 lim lim (x 2) lim f(x) 5 as in part (a). œ œ � Ê œ † ’ “ ’ “ x x x Ä # Ä # Ä # f(x) 5 x � � # 58. (a) 0 1 0 lim lim x lim lim x lim x lim f(x). That is, lim f(x œ œ œ œ œ † † ’ “ ’ “ ’ “ ’ “ ’ “ x x x x x x x Ä ! Ä ! Ä ! Ä ! Ä ! Ä ! Ä ! f(x) f(x) f(x) x x x # # # # # # ) 0. œ (b) 0 1 0 lim lim x lim x lim . That is, lim 0. œ œ œ œ œ † † ’ “ ’ “ ’ “ x x x x x Ä ! Ä ! Ä ! Ä ! Ä ! f(x) f(x) f(x) f(x) x x x x # # 59. (a) lim x sin 0 x Ä ! " x œ (b) 1 sin 1 for x 0: � Ÿ Ÿ Á " x x 0 x x sin x lim x sin 0 by the sandwich theorem; � Ê � Ÿ Ÿ Ê œ " " x x x Ä ! x 0 x x sin x lim x sin 0 by the sandwich theorem. � Ê � Ê œ " " x x x Ä ! 60. (a) lim x cos 0 x Ä ! # ˆ " x$ ‰ œ (b) 1 cos 1 for x 0 x x cos x lim x cos 0 by the sandwich � Ÿ Ÿ Á Ê � Ÿ Ÿ Ê œ ˆ ‰ ˆ ‰ ˆ ‰ " " " # # # # x x x $ $ $ x Ä ! theorem since lim x 0. x Ä ! # œ 2.3 PRECISE DEFINITION OF A LIMIT 1. Step 1: x 5 x 5 5 x 5 k � k � Ê � � � � Ê � � � � � $ $ $ $ $ Step 2: 5 7 2, or 5 1 4. $ $ $ $ � œ Ê œ � � œ Ê œ The value of which assures x 5 1 x 7 is the smaller value, 2. $ $ $ k � k � Ê � � œ Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley Section 2.3 Precise Definition of a Limit 79 2. Step 1: x 2 x 2 x 2 k � k � Ê � � � � Ê � � # � � � $ $ $ $ $ Step 2: 2 1 1, or 2 7 5. � � œ Ê œ � œ Ê œ $ $ $ $ The value of which assures x 2 1 x 7 is the smaller value, 1. $ $ $ k � k � Ê � � œ 3. Step 1: x ( 3) x 3 x 3 k k � � � Ê � � � $ � Ê � � � � � $ $ $ $ $ Step 2: 3 , or . � � œ � Ê œ � $ œ � Ê œ $ $ $ $ 7 5 # # # # " " The value of which assures x ( 3) x is the smaller value, . $ $ $ k k � � � Ê � � � � œ 7 # # # " " 4. Step 1: x x x ¸ ¸ ˆ � � ‰ � Ê � � � � Ê � � � � � 3 3 3 3 # # # # $ $ $ $ $ Step 2: , or 1. � � œ � Ê œ # � œ � Ê œ $ $ $ $ 3 7 3 # # # # " The value of which assures x x is the smaller value, . $ $ $ ¸ ¸ ˆ � � ‰ � Ê � � � � œ " 3 7 # # # " 5. Step 1: x x x ¸ ¸ � � Ê � � � � Ê � � � � � " " " " # # # # $ $ $ $ $ Step 2: , or . � � œ Ê œ � œ Ê œ $ $ $ $ " " " " # # 4 4 9 18 7 14 The value of which assures x x is the smaller value, . $ $ $ ¸ ¸ � � Ê � � œ " " # 4 4 9 7 18 6. Step 1: x 3 x 3 3 x 3 k � k � Ê � � � � Ê � � � � � $ $ $ $ $ Step 2: 2.7591 0.2409, or 3.2391 0.2391. � � $ œ Ê œ � $ œ Ê œ $ $ $ $ The value of which assures x 3 2.7591 x 3.2391 is the smaller value, 0.2391. $ $ $ k � k � Ê � � œ 7. Step 1: x 5 x 5 5 x 5 k � k � Ê � � � � Ê � � � � � $ $ $ $ $ Step 2: From the graph, 5 4.9 0.1, or 5 5.1 0.1; thus 0.1 in either case. � � œ Ê œ � œ Ê œ œ $ $ $ $ $ 8. Step 1: x ( 3) x 3 3 x 3 k k � � � Ê � � � � Ê � � � � � $ $ $ $ $ Step 2: From the graph, 3 3.1 0.1, or 3 2.9 0.1; thus 0.1. � � œ � Ê œ � œ � Ê œ œ $ $ $ $ $ 9. Step 1: x 1 x 1 1 x 1 k � k � Ê � � � � Ê � � � � � $ $ $ $ $ Step 2: From the graph, 1 , or 1 ; thus . � � œ Ê œ � œ Ê œ œ $ $ $ $ $ 9 7 25 9 7 16 16 16 16 16 10. Step 1: x 3 x 3 3 x 3 k � k � Ê � � � � Ê � � � � � $ $ $ $ $ Step 2: From the graph, 3 2.61 0.39, or 3 3.41 0.41; thus 0.39. � � œ Ê œ � œ Ê œ œ $ $ $ $ $ 11. Step 1: x 2 x 2 2 x 2 k � k � Ê � � � � Ê � � � � � $ $ $ $ $ Step 2: From the graph, 2 3 2 3 0.2679, or 2 5 5 2 0.2361; � � œ Ê œ � ¸ � œ Ê œ � ¸ $ $ $ $ È È È È thus 5 2. $ œ � È Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley 80 Chapter 2 Limits and Continuity 12. Step 1: x ( 1) x 1 1 x 1 k k � � � Ê � � � � Ê � � � � � $ $ $ $ $ Step 2: From the graph, 1 0.1180, or 1 0.1340; � � œ � Ê œ ¸ � œ � Ê œ ¸ $ $ $ $ È È È È 5 5 2 3 2 3 # # # # � � thus . $ œ È5 � 2 # 13. Step 1: x ( 1) x 1 1 x 1 k k � � � Ê � � � � Ê � � � � � $ $ $ $ $ Step 2: From the graph, 1 0.77, or 1 0.36; thus 0.36. � � œ � Ê œ ¸ � œ � Ê œ œ œ $ $ $ $ 16 7 16 9 9 9 9 25 25 25 14. Step 1: x x x ¸ ¸ � � Ê � � � � Ê � � � � � " " " " # # # # $ $ $ $ $ Step 2: From the graph, 0.00248, or 0.00251; � � œ Ê œ � ¸ � œ Ê œ � ¸ $ $ $ $ " " " " # # # # 1 1 1 1 2.01 2 .01 1.99 1.99 thus 0.00248. $ œ 15. Step 1: (x 1) 5 0.01 x 4 0.01 0.01 x 4 0.01 3.99 x 4.01 k k k k � � � Ê � � Ê � � � � Ê � � Step 2: x 4 x 4 4 x 4 0.01. k � k � Ê � � � � Ê � � � � � Ê œ $ $ $ $ $ $ 16. Step 1: (2x 2) ( 6) 0.02 2x 4 0.02 0.02 2x 4 0.02 4.02 2x 3.98 k k k k � � � � Ê � � Ê � � � � Ê � � � � 2.01 x 1.99 Ê � � � � Step 2: x ( 2) x 2 2 x 2 0.01. k k � � � Ê � � � � Ê � � � � � Ê œ $ $ $ $ $ $ 17. Step 1: x 1 0.1 0.1 x 1 0.1 0.9 x 1 1.1 0.81 x 1 1.21 ¹ ¹ È È È � � " � Ê � � � � " � Ê � � � Ê � � � 0.19 x 0.21 Ê � � � Step 2: x 0 x . Then, or ; thus, 0.19. k � k � Ê � � � � œ �!Þ"* Ê œ !Þ"* œ !Þ#" œ $ $ $ $ $ $ $ 18. Step 1: x 0.1 0.1 x 0.1 0.4 x 0.6 0.16 x 0.36 ¸ ¸ È È È � � Ê � � � � Ê � � Ê � � " " # # Step 2: x x . ¸ ¸ � � Ê � � � � Ê � � � B � � " " " " 4 4 4 4 $ $ $ $ $ Then, 0.16 0.09 or 0.36 0.11; thus 0.09. � � œ Ê œ � œ Ê œ œ $ $ $ $ $ " " 4 4 19. Step 1: 19 x 19 x 1 2 19 x 4 19 x 16 ¹ ¹ È È È � � $ � " Ê �" � � � $ � Ê � � � % Ê � � � x 19 16 15 x 3 or 3 x 15 Ê �% � � � � Ê � � � � Step 2: x 10 x 10 10 x 10. k k � � Ê � � � � Ê � � � � � $ $ $ $ $ Then 10 3 7, or 10 15 5; thus 5. � � œ Ê œ � œ Ê œ œ $ $ $ $ $ 20. Step 1: x 7 4 1 x 7 1 3 x 7 5 9 x 7 25 16 x 32 ¹ ¹ È È È � � � Ê �" � � � % � Ê � � � Ê � � � Ê � � Step 2: x 23 x 23 23 x 23. k k � � Ê � � � � Ê � � � � � $ $ $ $ $ Then 23 16 7, or 23 32 9; thus 7. � � œ Ê œ � œ Ê œ œ $ $ $ $ $ 21. Step 1: 0.05 0.05 0.05 0.2 0.3 x or x 5. ¸ ¸ " " " " " # x 4 x 4 x 3 3 10 10 10 � � Ê � � � � Ê � � Ê � � � � Step 2: x 4 x 4 4 x 4. k � k � Ê � � � � Ê � � � � � $ $ $ $ $ Then or , or 4 5 or 1; thus . � � % œ œ � œ œ œ $ $ $ $ $ 10 2 2 3 3 3 22. Step 1: x 3 .1 0.1 x 3 0.1 2.9 x 3.1 2.9 x 3.1 k k È È # # # � � ! Ê � � � � Ê � � Ê � � Step 2: x 3 x 3 3 x 3. ¹ È ¹ È È È � � Ê � � � � Ê � � � � � $ $ $ $ $ Then 3 2.9 3 2.9 0.0291, or 3 3.1 3.1 3 0.0286; � � œ Ê œ � ¸ � œ Ê œ � ¸ $ $ $ $ È È È È È È È È thus 0.0286. $ œ Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley Section 2.3 Precise Definition of a Limit 81 23. Step 1: x 4 0.5 0.5 x 4 0.5 3.5 x 4.5 3.5 x 4.5 4.5 x 3.5, k k k k È È È È # # # � � Ê � � � � Ê � � Ê � � Ê � � � � for x near 2. � Step 2: x ( 2) x 2 x 2. k k � � � Ê � � � � Ê � � # � � � $ $ $ $ $ Then 4.5 4.5 0.1213, or 3.5 3.5 0.1292; � � # œ � Ê œ � # ¸ � # œ � Ê œ # � ¸ $ $ $ $ È È È È thus 4.5 2 0.12. $ œ � ¸ È 24. Step 1: ( 1) 0.1 0.1 1 0.1 x or x . ¸ ¸ " " " x x 10 x 10 11 9 9 11 11 9 10 10 10 10 � � � Ê � � � � Ê � � � � Ê � � � � � � � � Step 2: x ( 1) x 1 x . k k � � � Ê � � � � Ê � � " � � � " $ $ $ $ $ Then , or ; thus . � � " œ � Ê œ � " œ � Ê œ œ $ $ $ $ $ 10 10 9 9 11 11 11 " " " 25. Step 1: x 5 11 x 16 1 x 16 1 15 x 17 15 x 17. k k k k a b È È # # # # � � � " Ê � � Ê �" � � � Ê � � Ê � � Step 2: x 4 x 4 x . k � k � Ê � � � � Ê � � % � � � % $ $ $ $ $ Then 15 15 0.1270, or 17 17 0.1231; � � % œ Ê œ % � ¸ � % œ Ê œ � % ¸ $ $ $ $ È È È È thus 17 4 0.12. $ œ � ¸ È 26. Step 1: 5 1 4 6 30 x 20 or 20 x 30. ¸ ¸ 120 120 120 x x x x 4 120 6 � � " Ê �" � � & � Ê � � Ê � � Ê � � � � " " Step 2: x 24 x 24 24 x 24. k k � � Ê � � � � Ê � � � � � $ $ $ $ $ Then 24 20 4, or 24 30 6; thus 4. � � œ Ê œ � œ Ê œ Ê œ $ $ $ $ $ 27. Step 1: mx 2m 0.03 0.03 mx 2m 0.03 0.03 2m mx 0.03 2m k k � � Ê � � � � Ê � � � � � Ê 2 x 2 . � � � � 0.03 0.03 m m Step 2: x 2 x 2 x . k � k � Ê � � � � Ê � � # � � � # $ $ $ $ $ Then , or . In either case, . � � # œ # � Ê œ � # œ # � Ê œ œ $ $ $ $ $ 0.03 0.03 0.03 0.03 0.03 m m m m m 28. Step 1: mx 3m c c mx 3m c c 3m mx c 3m 3 x 3 k k � � Ê � � � � Ê � � � � � Ê � � � � c c m m Step 2: x 3 x 3 . k � k � Ê � � � � Ê � � $ � B � � $ $ $ $ $ $ Then , or . In either case, . � � $ œ $ � Ê œ � $ œ $ � Ê œ œ $ $ $ $ $ c c c c c m m m m m 29. Step 1: (mx b) b c mx c c mx c x . ¸ ¸ ˆ ‰ � � � � - Ê � � � � Ê � � � � � Ê � � � � m m m m c c m m # # # # # # " " Step 2: x x x . ¸ ¸ � � Ê � � � � Ê � � � � � " " " " # # # # $ $ $ $ $ Then , or . In either case, . � � œ � Ê œ � œ � Ê œ œ $ $ $ $ $ " " " " # # # # c c c c c m m m m m 30. Step 1: (mx b) (m b) 0.05 0.05 mx m 0.05 0.05 m mx 0.05 m k k � � � � Ê � � � � Ê � � � � � 1 x . Ê � � � " � 0.05 0.05 m m Step 2: x 1 x 1 x . k � k � Ê � � � � Ê � � " � � � " $ $ $ $ $ Then , or . In either case, . � � " œ " � Ê œ � " œ " � Ê œ œ $ $ $ $ $ 0.05 0.05 0.05 0.05 0.05 m m m m m 31. lim (3 2x) 3 2(3) 3 x Ä 3 � œ � œ � Step 1: 3 2x ( 3) 0.02 0.02 6 2x 0.02 6.02 2x 5.98 3.01 x 2.99 or k k a b � � � � Ê � � � � Ê � � � � � Ê � � 2.99 x 3.01. � � Step 2: 0 x 3 x 3 x . � � � Ê � � � � Ê � � $ � � � $ k k $ $ $ $ $ Then 2.99 0.01, or 3.01 0.01; thus 0.01. � � $ œ Ê œ � $ œ Ê œ œ $ $ $ $ $ 32. lim ( 3x ) ( 3)( 1) 2 1 x Ä �1 � � # œ � � � œ Step 1: ( 3x 2) 1 0.03 0.03 3x 3 0.03 0.01 x 1 0.01 1.01 x 0.99. k k � � � � Ê � � � � � Ê � � � � Ê � � � � Copyright (c) 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley Copy