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a lim d x lim x 4 1000000x 3, b Big positive wins since x 4 gets larger a lot quicker than x 3. a The dominant term is 2y 5 b The dominant term is 20x 9. c The dominant term is x 6 d The dominant term is x. Review and Preview 8 1 1,a lim x 6 b lim x 6 12x 5 7x 3 400. c lim 3x 7 d lim 3x 7 1000x 6 48 000,a lim 50 log x x 2 Dominant term is x 2. b lim 1 5 x 300x 4 Dominant term is 1 5 x,c lim 1000 2 x 3x Dominant term is 3x. d lim x 30 x Dominant term is x,a See graph at right.
b 4 4 2 4 12 2 2 4 6 16 10 24 50,c 50 6 10 50 60 110. a 2 3 x 2 2 2 x b 200 1 05 x 5000,3x 4 2x 1 05 x 25. 5x 4 log1 05 1 05 x log1 05 25,Solution continues on next page. CPM Educational Program 2012 Chapter 8 Page 2 Pre Calculus with Trigonometry. 8 12 Solution continued from previous page,c log 3 x 5 4 d 20m1 5 1000. 3log 3 x 5 34 m1 5 50,m 3 2 2 3 502 3,a Add 4 b Divide by 2 c Add 1.
slope 23 slope 3,y 5 23 x 7,2y 10 3x 21, a y 10x vertical asymptote at x 0 horizontal asymptote at y 0. b y 0 5 x no vertical asymptote horizontal asymptote at y 0. vertical asymptote at x 2 horizontal asymptote at y 0. d y 2 x 3 no vertical asymptote horizontal asymptote at y 3. a 3 f x 3 1x 20 3x 60 b 2 f x 2 1x 20 2x 40,5 x 5x 5x x15 20 1 5x 20 5x 20. 1 f x 1 1 20 1 4,sin 2 x cos2 x sin 2 x cos2 x 1,sin 2 x cos2 x 1. 1 cos2 x cos2 x 1,2 cos2 x 1, CPM Educational Program 2012 Chapter 8 Page 3 Pre Calculus with Trigonometry. Lesson 8 1 2,The limits for all of the three functions are 2.
a Answers may vary but students will probably find iii the easiest to evaluate since they. should remember little over big goes to zero leaving 21. 2x 3 x 2 x2 x2 2x 1,x 3x x 3x 3 1 3,x 2x 2 2 x, To change into x 7 divide the numerator and denominator by x 3. 3x 4 2 3 2,3x 4 2 x4 x4 4 3,a lim lim lim x2 0 5,x 6x 4 2x x 6 x4 2 x x 6 6. 3x 4 2 3 2,3x 4 2 5 5 x x5 0,b lim lim x 5 x lim 0. x 6x 5 2x x 6 x 2 x x 6 2 6,3x 4 2 3x 2,3x 4 2 x3 x3 x3 3x. c lim lim lim lim,x 6x 3 2x x 6 x3 2 x x 6 2 x 6,a The limit is zero 0.
b The limit will be or, c The limit will be the ratio of the coefficient of the dominant terms. a lim 8 x 170 x 3 lim x 3 8 x 170,x 50 x 2 17 x 100 x 50 x 2 17 x 100. b lim 5 x 124x 103 lim 5 x4 123x 10 13,x 18 x 15 x 3x x 15 x 3x 18 x. 2 x 7 3 4 x 2 28 x 49 2 x 7 8 x 3 28 x 2 56 x 2 196 x 98 x 343. c lim lim lim 8, x 3x 3 12 x 10 x 2 x 3x 3 12 x 10 x 2 x 3x 3 12 x 10 x 2 3. CPM Educational Program 2012 Chapter 8 Page 4 Pre Calculus with Trigonometry. a lim 2 x 8 lim 2 x 2 8 Limit 0 Dominant terms numerator 2 x denominator 3x. b lim 10x 6, Limit Dominant terms numerator x denominator log x.
log 5 x x log 5 log 5, c lim lim Dominant terms numerator x log 5 denominator 2x. x 2 x 3 x 2 x 3 2,x 4 3x 1000, d lim 1 Dominant terms numerator x 4 x 2 denominator 2x 2 4x 2. x 2 x 3 2 4,Review and Preview 8 1 2,lim 3 lim 2 0. x x x x x x x,x 2 5 x2 x2,c The lim cos x does not exist. d lim x 1 lim x,1 therefore lim does not exist,x 0 x 0 x 0 x.
e lim log 2 x, a It approaches zero since the dominant numerator is x and the dominant denominator is x 2. b We have a horizontal asymptote at y 0,c lim f x and lim f x. d There is an error at x 1 and this is reflected on the graph by a vertical asymptote. a lim f x 1,3 2 6 3 10,d lim f x 37,x 3 3 3 2 36,e The function is undefined at x 3. f The function s value is 37, CPM Educational Program 2012 Chapter 8 Page 5 Pre Calculus with Trigonometry. a x2 4 5 10 10 b y2 0 2 4 4,x3 4 10 10 40 10 30 y3 4 2 4 16 4 12.
x4 4 30 10 120 10 110 y4 12 2 4 144 4 140,x5 4 110 10 440 10 430 y5 140 2 4 19 596. d w2 3 5 15,1 41 4 w3 5 15 75,w4 15 75 1125,w5 75 1125 84 375. t 2 5 2 1 7 t 3 7 2 2 11 t 4 11 2 3 17 t 5 17 2 4 25. The end behavior of the graph, b lim 3x 6 2 x 3x is the dominant term in the function. d lim 5 x 10 4 x 5 x is the dominant term in the function. The larger base will determine if the function goes to or. CPM Educational Program 2012 Chapter 8 Page 6 Pre Calculus with Trigonometry. Lesson 8 1 3, a f x x 1 g x x 2 these values would cause division by zero. x 5 4 3 2 1 0 1 2 3 4 5,f x 1 5 0 67 0 0 und 6 6 6 67 7 5 8 4 9 33.
x 5 4 3 2 1 0 1 2 3 4 5,g x 2 1 0 und 2 3 4 5 6 7 8. d In the table y error The function is not defined There appears to be a vertical asymptote. for one of the graphs but the other looks linear, They have similar form domain is all real numbers except one point. f x has a vertical asymptote g x has a hole Common factor in numerator and. denominator creates a hole in the graph of a rational function. x 2 4 x 2 x 2,y1 x 2 x 2,Hole at x 2,2 x 6 Vertical asymptotes at x 3 and x 4. Horizontal asymptote at y 0,y3 4x Horizontal asymptote at y 0. They simplify to the same function but have different discontinuities f x x. vertical asymptote at x 2,g x x 2 2 x,x looks like f x but has a hole at x 2.
x2 4 x 2 x 2 x 2, CPM Educational Program 2012 Chapter 8 Page 7 Pre Calculus with Trigonometry. Answers will vary Examples are given,1 x 2 9 x3 x,x 2 2 x 24 x 2 x 2 x 2 9. Review and Preview 8 1 3,a x2 3 x2 x2 3 x2 1 3 x2,lim lim lim 1. 2 x 2x x 5x x,2 2 2 x 2 5 x 2,5x 3 7x 5x 3 x 3 7x x 3 5 7 x2. b lim lim lim 0,4 x x x 4x x,4 3 3 x x 4 x 2,c 6x 3 8x 2 6x 3 x 2 8x 2 x 2 6x 8.
lim lim lim,2 x 15x x 2 x,2 2 2 x 15 2 x 2,5 x 2 2 x 3 0 2 x3 7. a lim b lim 2,x x 3 1 x 3x 3 4 x 5 3,c lim 23x 7 3 7. x 0 3x 4 x 5 3 0 4 0 5,lim g x lim 3x x,lim x lim 3 The limit equals 3. x x x 2 5 x x 2 x 2 5 x 2 x 1 5 x 2,a lim h x 2 lim 3x x. b lim j x lim log 3x x,c See graphs at right, CPM Educational Program 2012 Chapter 8 Page 8 Pre Calculus with Trigonometry.
a m x log log 2 log x x log 2 5 log x 0 301x 5 log x. b lim k x lim The dominant term is 2,lim m x lim log. lim log 2 lim log x,a Area of a side L2 Surface area of a cube S 6L2. b Length of the diagonal on the face of the cube D12 L2 L2. Length of the longest diagonal D 2 L2 2L 2,D 2 L2 2L2. D S 6 3 3S 6 S 2, a Initial value 1 87 multiplier 1 0 02 1 02 time is in months. b Initial value 12 000 multiplier 1 0 12 0 88 time is in years. c Initial value 1 000 multiplier 1 0 30 1 30 time is in hours. d Initial value 5 000 multiplier 1 0 01 1 01 time is in months. a y 5000 1 0112 b y 5000 1 02 4,y 5 634 13 y 5 412 16.
c y 5000 1 00152 d y 5000 1 0002 365,y 5 266 74 y 5 378 61. CPM Educational Program 2012 Chapter 8 Page 9 Pre Calculus with Trigonometry. Lesson 8 1 4,b cos 15 h1 h cos 15,c sin 15 1,2 sin 15 b. one triangle A 12 2 sin 15 cos 15 sin 15 cos 15,Area of polygon A 12 sin 15 cos 15 3. a tan 15 1,b 2 tan 15 b,b Area of one triangle A 12 2 tan 15 1 tan 15. Area of polygon A 12 tan 15 3 215,c 3 and 3 215,c cos 360 2.
n d 2 sin 360 2,n 2 sin 180,e n f n sin 180 n cos 180 n n2 sin 360. a The smallest number of sides of a polygon is 3, c The plot shows the areas of the inscribed polygons as n increases. a n tan 360, d The value of gets squeezed between the two functions which act as an upper bound and. a lower bound, CPM Educational Program 2012 Chapter 8 Page 10 Pre Calculus with Trigonometry. Review and Preview 8 1 4,lim 1 lim 1 lim,1n 1 0 1 b lim 1 01.
a 1050 1 0175 20 b 5000 1 082,c P 1 100n,a First four terms b First four terms. n 1 2 n 2 1 1 n 1 23,3 n 3 1 4 3n 31,2 2 0 n 2 49,n 3 2 n 2 3 1 2 n 2 3. 3 n 3 3 6 n 3 27 8,lim 2 n nn 1 2 n,n 3 n lim 0,c First four terms. There is no limit for lim n,See graph at right,a lim g x 2. c g x 3 2 x,a Increasing b Decreasing c Decreasing.
CPM Educational Program 2012 Chapter 8 Page 11 Pre Calculus with Trigonometry. logb RQ logb T,logb R logb Q logb T 1,5 4 1 2 1 4 1. Center of the circle 0 0 Slope from center of circle to point 3 4 m 4 0. Equation of the line y 4 43 x 3 or y 43 x 3 4,a See graph at right below. b y 3 0 01t,c Area under the curve after 1 minute A 1. 2 3 3 0 01 60 60 198 feet, Area under the curve after 5 minutes A 12 3 3 0 01 300 300 1350 feet. A 12 3 3 0 01T T 3T 0 005T 2 v t,velocity ft sec,Lesson 8 2 1 t.
100 200 300, goes to zero so the expression in the parentheses goes to 1 and the limit might be 1. since 1n 1, b Infinity since the number inside the parentheses is larger than 1. c A number around 2 718 It should be surprising that this limit is neither 1 nor infinity. 2 718281828,a e3 b e0 2,0 064 0 0612,a 1000 1 b 1000 1. 1000 1 0 06 1000 1 24 365,365 0 06 24 365,e 1000e0 06 f a 1061 36. g 1200e0 075 1293 46 b 10061 68,e more and more, CPM Educational Program 2012 Chapter 8 Page 12 Pre Calculus with Trigonometry.
a y log 2 x b y log e x,See graph at right below,a log 3 log 3. b log 3 54 log 3 2 log 3 542 log 3 27 log 3 33 3,ln e12 ln e e12 12 ln e e 12. 5 ln e2 10 ln e 10,ln 1 e ln e 1 ln e, CPM Educational Program 2012 Chapter 8 Page 13 Pre Calculus with Trigonometry. Review and Preview 8 2 1,17 32 58 e x 3,0 299 e x e3. 0 01489 e x,ln 0 01489 x,a k1 01 1 1 21 3 1 41 1 1 12 16 241 2 708333.
b k1 01 1 1 21 3 1 41 51 61 71 81 91 101 2 71828,1 3 7 1 6 x 3 7 x 6 x 1 x 3 0x 2 7x 6. 20 which is undefined x 1,b See division at right x 3 x 2. 3 7 x 6 x 2 7x,lim x x 1 lim x 2 x 6 1 2 1 6 4,x 1 x 1 x 2 1x. c Hole at x 1 6x 6,See graph at right y,Point of intersection 1 7. CPM Educational Program 2012 Chapter 8 Page 14 Pre Calculus with Trigonometry. t 2 1 2 1 1 4,t 3 4 2 2 1 9,t 4 9 2 3 1 16,t 5 16 2 4 1 25.
Let a x 3 If ax 2 0 If x 2 a 3 0,x 3a 4 2x 2 a 3 0 x 3 x 2 0 x 2 x 3 3 0. x 2 a 3 ax 2 0 x 2 3x 2 0 x 3 0,x 2 a 3 0 or ax 2 0 x 2 x 1 0. x intercepts 0 32t 16t 2 16t 2 t t 0 2,Vertex avg of x intercepts or t 1. At t 1 s 32 1 16 12 16 ft,cot opposite,f x 2 x 2 3x 2. x2 4 x 2 x 2 x 2,Hole at x 2 f 2,Asymptotes at x 2 and y 2.
Lesson 8 2 2,a A 100e0 065 1 b A 100e0 065 2,A 106 72 A 113 89. c A 100e0 065 3 d A 100 e0 06 17,A 320 15 A 301 92. e A 100 e0 06 t, CPM Educational Program 2012 Chapter 8 Page 15 Pre Calculus with Trigonometry. for 2 1 2 gt