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1. f(x)= 3. f(x)= 1-2r 2²-1¹ x-4 x²-4x+41 a = 1 x² + 2x-1 5. lim 7. lim cotx. 1-40 2²-4 1-2x 2. f(x)= 4. f(x)= a=2 1-x (x + 1)2² In exercises 5-22, determine each limit (answer as appropriate, with a number, ∞, -o or does not exist). a=-1 a=-1 6. lim (x²-2x-3)-2/3 8. lim xsec³ x

1. f(x)= 3. f(x)= 1-2r 2²-1¹ x-4 x²-4x+41 a = 1 x² + 2x-1 5. lim 7. lim cotx. 1-40 2²-4 1-2x 2. f(x)= 4. f(x)= a=2 1-x (x + 1)2² In exercises 5-22, determine each limit (answer as appropriate, with a number, ∞, -o or does not exist). a=-1 a=-1 6. lim (x²-2x-3)-2/3 8. lim xsec³ x

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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EXERCISES 1.5 are never crossed. Explain why horizontal or slant asymp- totes may, in fact, be crossed any number of times: draw one example. NWRITING EXERCISES 1. It may seem odd that we use o in describing limits but do not count oo as a real number. Discuss the existence of oc: is it a number or a concept? 2. In example 5.7, we dealt with the "indeterminate form" Thinking of a limit of oo as meaning "getting very large" and a limit of 0 as meaning "getting very close to 0," explain why the following are indeterminate forms: , . 00 - oo, and 00 - 0. Determine what the following non-indeterminate forms represent: 00 + 0o, -00 - oo, 00 +0 and 0/0o. In exercises 1-4, determine (a) lim f(x) (b) lim f(x) and (c) lim f(x) (answer as appropriate, with a number, oo,-00 or does not exist). 1- 2r a =1 x -1 1- 2x 1. fx) = 2. fx) = a = -I x2 -1 3. On your computer or calculator, graph y = 1/(r - 2) and look for the horizontal asymptote y = 0 and the vertical asymptote * = 2. Many computers will draw a vertical line at x = 2 and will show the graph completely flattening out at y = 0 for large x's. Is this accurate? misleading? Most computers will compute the locations of points for adjacent x's and try to connect the points with a line segment. Why might this result in a vertical line at the location of a vertical asymptote? x-4 1-x 3. f(x) = - 4x +4 a=2 4. f) = r+ 1 In exercises 5-22, determine each limit (answer as appropriate, with a number, oo,-00 or does not exist). x +2x - 1 x-4 7. lim cot x 5. lim 6. lim (x - 2x - 3)2/3 4. Many students learn that asymptotes are lines that the graph gets closer and closer to without ever reaching. This is true for many asymptotes, but not all. Explain why vertical asymptotes 8. lim x sec'x /2 104 CHAPTER I Limits and Continuity 1-40 0.00016 C/Users/96658/Desktop/redh/Smith - Calculus_Early_Transcende juai + - 104 CHAPTER I . Limits and Continuity 1-40 x2+ 3x - 2 2x -x+1 A 38. Graph the velocity function in exercise 37 with k = 0.00016 (representing a headfirst dive) and estimate how long it takes for the diver to reach a speed equal to 90% of the limiting velocity. Repeat with k = 0.001 (representing a spread-eagle position). 9. lim 3x2 + 4x - 1 10. lim 1 4x - 3r - 1 2x -1 11. lim 12. lim I 4x - 5x - 1 - V4 +x² 13. lim In 14. lim In(x sin x) In exercises 39-48, use graphical and numerical evidence to conjecture a value for the indicated limit. 15. lim e2/ 16. lim etr+l+2 In(x + 2) In(2 +) 39. lim 40, lim In(x + 3x + 3) In(1+ er) 17. lim cot x 18. lim sec x +1 - 4x + 7 21 +7x+1 19. lim sin(e-1/ 41. lim 2x2 +x cos x 42. lim 20. lim sin(tanx) x -x sinx x+ 4x +5 21. lim e tan 22. lim tan(In x) 43. lim 44. lim (e/ -x*) e/2 In x 46. lim e - 1 45. lim In exercises 23-28, determine all horizontal and vertical asymp- 47. lim xh 48. lim x totes. For each side of each vertical asymptote, determine whether f(x) - 0o or f(x) - -0o. 23. (a) f(x) =- 4-x (b) f(x) = 4-x In exercises 49 and 50, use graphical and numerical evidence to conjecture the value of the limit. Then, verify your conjecture by finding the limit exactly. 24. (a) f(x) = (b) f(x) = V4+x V4 -x2 3x +1 25. f(x)= - 2x -3 1-x 26. f(x) = +x- 2 49. lim (V4x? - 2x +1- 2r) (Hint: Multiply and divide by the 27. f(x) = 4 tanx - 1 28. f(x) = In(1 - cos x) conjugate expression: 4x - 2x +I+2x and simplify.) 50. lim (v5x? + 4x +7- V5x? +x + 3) (See the hint for
Transcribed Image Text:EXERCISES 1.5 are never crossed. Explain why horizontal or slant asymp- totes may, in fact, be crossed any number of times: draw one example. NWRITING EXERCISES 1. It may seem odd that we use o in describing limits but do not count oo as a real number. Discuss the existence of oc: is it a number or a concept? 2. In example 5.7, we dealt with the "indeterminate form" Thinking of a limit of oo as meaning "getting very large" and a limit of 0 as meaning "getting very close to 0," explain why the following are indeterminate forms: , . 00 - oo, and 00 - 0. Determine what the following non-indeterminate forms represent: 00 + 0o, -00 - oo, 00 +0 and 0/0o. In exercises 1-4, determine (a) lim f(x) (b) lim f(x) and (c) lim f(x) (answer as appropriate, with a number, oo,-00 or does not exist). 1- 2r a =1 x -1 1- 2x 1. fx) = 2. fx) = a = -I x2 -1 3. On your computer or calculator, graph y = 1/(r - 2) and look for the horizontal asymptote y = 0 and the vertical asymptote * = 2. Many computers will draw a vertical line at x = 2 and will show the graph completely flattening out at y = 0 for large x's. Is this accurate? misleading? Most computers will compute the locations of points for adjacent x's and try to connect the points with a line segment. Why might this result in a vertical line at the location of a vertical asymptote? x-4 1-x 3. f(x) = - 4x +4 a=2 4. f) = r+ 1 In exercises 5-22, determine each limit (answer as appropriate, with a number, oo,-00 or does not exist). x +2x - 1 x-4 7. lim cot x 5. lim 6. lim (x - 2x - 3)2/3 4. Many students learn that asymptotes are lines that the graph gets closer and closer to without ever reaching. This is true for many asymptotes, but not all. Explain why vertical asymptotes 8. lim x sec'x /2 104 CHAPTER I Limits and Continuity 1-40 0.00016 C/Users/96658/Desktop/redh/Smith - Calculus_Early_Transcende juai + - 104 CHAPTER I . Limits and Continuity 1-40 x2+ 3x - 2 2x -x+1 A 38. Graph the velocity function in exercise 37 with k = 0.00016 (representing a headfirst dive) and estimate how long it takes for the diver to reach a speed equal to 90% of the limiting velocity. Repeat with k = 0.001 (representing a spread-eagle position). 9. lim 3x2 + 4x - 1 10. lim 1 4x - 3r - 1 2x -1 11. lim 12. lim I 4x - 5x - 1 - V4 +x² 13. lim In 14. lim In(x sin x) In exercises 39-48, use graphical and numerical evidence to conjecture a value for the indicated limit. 15. lim e2/ 16. lim etr+l+2 In(x + 2) In(2 +) 39. lim 40, lim In(x + 3x + 3) In(1+ er) 17. lim cot x 18. lim sec x +1 - 4x + 7 21 +7x+1 19. lim sin(e-1/ 41. lim 2x2 +x cos x 42. lim 20. lim sin(tanx) x -x sinx x+ 4x +5 21. lim e tan 22. lim tan(In x) 43. lim 44. lim (e/ -x*) e/2 In x 46. lim e - 1 45. lim In exercises 23-28, determine all horizontal and vertical asymp- 47. lim xh 48. lim x totes. For each side of each vertical asymptote, determine whether f(x) - 0o or f(x) - -0o. 23. (a) f(x) =- 4-x (b) f(x) = 4-x In exercises 49 and 50, use graphical and numerical evidence to conjecture the value of the limit. Then, verify your conjecture by finding the limit exactly. 24. (a) f(x) = (b) f(x) = V4+x V4 -x2 3x +1 25. f(x)= - 2x -3 1-x 26. f(x) = +x- 2 49. lim (V4x? - 2x +1- 2r) (Hint: Multiply and divide by the 27. f(x) = 4 tanx - 1 28. f(x) = In(1 - cos x) conjugate expression: 4x - 2x +I+2x and simplify.) 50. lim (v5x? + 4x +7- V5x? +x + 3) (See the hint for
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