Problem 1E: Explain with examples what is meant by the indeterminate form 0/0. Problem 2E: Why are special methods, such as lHpitals Rule, needed to evaluate indeterminate forms (as opposed... Problem 3E: Explain the steps used to apply lHpitals Rule to a limit of the form 0/0. Problem 4E Problem 5E: Explain how to convert a limit of the form 0 to a limit of the form 0/0 or /. Problem 6E: Give an example of a limit of the form / as x 0. Problem 7E: Explain why the form 1 is indeterminate and cannot be evaluated by substitution. Explain how the... Problem 8E: Give the two-step method for attacking an indeterminate limit of the form limxaf(x)g(x). Problem 9E: In terms of limits, what does it mean for f to grow faster than g as x ? Problem 10E: In terms of limits, what does it mean for the rates of growth of f and g to be comparable as x ? Problem 11E: Rank the functions x3, ln x, xx, and 2x in order of increasing growth rates as x . Problem 12E: Rank the functions x100, ln x10, xx, and 10x in order of increasing growth rates as x . Problem 13E: 0/0 form Evaluate the following limits using lHpitals Rule. 13. limx2x22x86x+x2 Problem 14E: 0/0 form Evaluate the following limits using lHpitals Rule. 14. limx1x4+x3+2x+2x+1 Problem 15E: 0/0 form Evaluate the following limits using lHpitals Rule. 15. limx1lnx4xx23 Problem 16E: 0/0 form Evaluate the following limits using l'Hpitals Rule. 16. limx0ex1x2+3x Problem 17E: 0/0 form Evaluate the following limits using lHpitals Rule. 17. limxelnx1xe Problem 18E: 0/0 form Evaluate the following limits using lHpitals Rule. 18. limx14tan1xx1 Problem 19E Problem 20E: 0/0 form Evaluate the following limits using l'Hpitals Rule. 20. limx2xsinx+x242x2 Problem 21E: 0/0 form Evaluate the following limits using lHpitals Rule. 21. limutanucotuu/4 Problem 22E: 0/0 form Evaluate the following limits using lHpitals Rule. 22. limz0tan4ztan7z Problem 23E: 0/0 form Evaluate the following limits. 23. limx01cos3x8x2 Problem 24E: 0/0 form Evaluate the following limits. 24. limx0sin23xx2 Problem 25E: 0/0 form Evaluate the following limits. 25. limxcosx+1(x)2 Problem 26E: 0/0 form Evaluate the following limits. 26. limx0=exx15x2 Problem 27E: 0/0 form Evaluate the following limits. 27. limx0exsinx1x4+8x3+12x2 Problem 28E: 0/0 form Evaluate the following limits. 28. limx0sinxx7x3 Problem 29E Problem 30E: 0/0 form Evaluate the following limits. 30. limxtan1x/21/x Problem 31E: 0/0 form Evaluate the following limits. 31. limx1x3x25x3x4+2x3x24x2 Problem 32E: 0/0 form Evaluate the following limits. 32. limx1xn1x1, n is a positive integer Problem 33E: 0/0 form Evaluate the following limits. 33. limx3v1v25v3 Problem 34E: 0/0 form Evaluate the following limits. 34. limy2y2+y68y2y Problem 35E: 0/0 form Evaluate the following limits. 35. limx2x24x+4sin2(x) Problem 36E: 0/0 form Evaluate the following limits. 36. limx23x+232x2 Problem 37E Problem 38E: / form Evaluate the following limits. 38. limx4x32x2+6x3+4 Problem 39E: / form Evaluate the following limits. 39. limx/2tanx3/(2x) Problem 40E Problem 41E: / form Evaluate the following limits. 41. limxln(3x+5)ln(7x+3)+1 Problem 42E: / form Evaluate the following limits. 42. limxln(3x+5ex)ln(7x+3e2x) Problem 43E Problem 44E Problem 45E: 0 form Evaluate the following limits. 45. limx0xcscx Problem 46E: 0 form Evaluate the following limits. 46. limx1(1x)tan(x2) Problem 47E: 0 form Evaluate the following limits. 47. limx0csc6xsin7x Problem 48E: 0 form Evaluate the following limits. 48. limx(csc(1/x)(e1/x1)) Problem 49E: 0 form Evaluate the following limits. 49. limx/2(2x)secx Problem 50E: 0 form Evaluate the following limits. 50. limx0+(sinx)1xx Problem 51E: form Evaluate the following limits. 51. limx0(cotx1x) Problem 52E Problem 53E: form Evaluate the following limits. 53. lim/2(tansec) Problem 54E Problem 55E Problem 56E Problem 57E Problem 58E: 1, 00, 0 forms Evaluate the following limits or explain why they do not exist. Check your results by... Problem 59E: 1, 00, 0 forms Evaluate the following limits or explain why they do not exist. Check your results by... Problem 60E Problem 61E Problem 62E Problem 63E: 1, 00, 0 forms Evaluate the following limits or explain why they do not exist. Check your results by... Problem 64E: 1, 00, 0 forms Evaluate the following limits or explain why they do not exist. Check your results by... Problem 65E Problem 66E: 1, 00, 0 forms Evaluate the following limits or explain why they do not exist. Check your results by... Problem 67E Problem 68E Problem 69E: Comparing growth rates Use limit methods to determine which of the two given functions grows faster... Problem 70E: Comparing growth rates Use limit methods to determine which of the two given functions grows faster... Problem 71E: Comparing growth rates Use limit methods to determine which of the two given functions grows faster... Problem 72E: Comparing growth rates Use limit methods to determine which of the two given functions grows faster... Problem 73E: Comparing growth rates Use limit methods to determine which of the two given functions grows faster... Problem 74E: Comparing growth rates Use limit methods to determine which of the two given functions grows faster... Problem 75E: Comparing growth rates Use limit methods to determine which of the two given functions grows faster... Problem 76E Problem 77E Problem 78E Problem 79E: Comparing growth rates Use limit methods to determine which of the two given functions grows faster... Problem 80E Problem 81E: Explain why or why not Determine whether the following statements are true and give an explanation... Problem 82E: Two methods Evaluate the following limits in two different ways: Use the methods of Chapter 2 and... Problem 83E: Two methods Evaluate the following limits in two different ways: Use the methods of Chapter 2 and... Problem 84E Problem 85E: Miscellaneous limits by any means Use analytical methods to evaluate the following limits. 85.... Problem 86E: Miscellaneous limits by any means Use analytical methods to evaluate the following limits. 86.... Problem 87E: Miscellaneous limits by any means Use analytical methods to evaluate the following limits. 87.... Problem 88E: Miscellaneous limits by any means Use analytical methods to evaluate the following limits. 88.... Problem 89E: Miscellaneous limits by any means Use analytical methods to evaluate the following limits. 89.... Problem 90E: Miscellaneous limits by any means Use analytical methods to evaluate the following limits. 90.... Problem 91E: Miscellaneous limits by any means Use analytical methods to evaluate the following limits. 91.... Problem 92E: Miscellaneous limits by any means Use analytical methods to evaluate the following limits. 92.... Problem 93E: Miscellaneous limits by any means Use analytical methods to evaluate the following limits. 93.... Problem 94E: Miscellaneous limits by any means Use analytical methods to evaluate the following limits. 94.... Problem 95E: Miscellaneous limits by any means Use analytical methods to evaluate the following limits. 95.... Problem 96E: Miscellaneous limits by any means Use analytical methods to evaluate the following limits. 96.... Problem 97E: Miscellaneous limits by any means Use analytical methods to evaluate the following limits. 97.... Problem 98E: Miscellaneous limits by any means Use analytical methods to evaluate the following limits. 98.... Problem 99E: Miscellaneous limits by any means Use analytical methods to evaluate the following limits. 99.... Problem 100E: Miscellaneous limits by any means Use analytical methods to evaluate the following limits. 100.... Problem 101E: Miscellaneous limits by any means Use analytical methods to evaluate the following limits. 101.... Problem 102E: Miscellaneous limits by any means Use analytical methods to evaluate the following limits. 102.... Problem 103E Problem 104E: Limits with parameters Evaluate the following limits in terms of the parameters a and b, which are... Problem 105E: Limits with parameters Evaluate the following limits in terms of the parameters a and b, which are... Problem 106E: Limits with parameters Evaluate the following limits in terms of the parameters a and b, which are... Problem 107E: Limits with parameters Evaluate the following limits in terms of the parameters a and b, which are... Problem 108E: An optics limit The theory of interference of coherent oscillators requires the limit... Problem 109E: Compound interest Suppose you make a deposit of P into a savings account that earns interest at a... Problem 110E: Algorithm complexity The complexity of a computer algorithm is the number of operations or steps the... Problem 111E: LHpital loops Consider the limit limx0ax+bcx+d, where a, b, c, and d are positive real numbers. Show... Problem 112E: General result Let a and b be positive real numbers. Evaluate limx(axa2x2bx) in terms of a and b. Problem 113E: Exponential functions and powers Show that any exponential function bx, for b 1, grows faster than... Problem 114E: Exponentials with different bases Show that f(x) = ax grows faster than g(x) = bx as x if 1 b a. Problem 115E: Logs with different bases Show that f(x) = loga x and g(x) = logb x, where a l and b 1. grow at a... Problem 116E: Factorial growth rate The factorial function is defined for positive integers as n!=n(n1)(n2)321.... Problem 117E: A geometric limit Let f() be the area of the triangle ABP (see figure) and let g() be the area of... Problem 118E Problem 119E Problem 120E: Exponentials vs. super exponentials Show that xx grows faster than bx as x , for b 1. Problem 121E: Exponential growth rates a. For what values of b 0 does bx grow faster than as ex as x ? b.... format_list_bulleted