Need help with a calculus problem about limits

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56. y Inx Va Applying l'Hôpital's Rule Use l'Hôpital's rule to find the limits in Exercises 7-50. 42. lim (cScx cotx + cos x) 44. lim el - (1 + h) ha Indeterminate Powers and Products Find the limits in Exercises 51-66. 60. lim 1 + X r0 (1+ 64. lim x(In x)? r0 Theory and Applications L'Hôpital's Rule does not help with the limits in Exercises 67-74. Try it-you just keep on cycling. Find the limits some other way. V9x + 1 Va 67. lim 68. lim +00 V x + 1 0 V sinx secx 69. lim 70. lim cotx CSC X 1/2) tani 10 2* – 34 71. lim 2* + 4 72. lim +34 + 4* --0054 - 24 80. For what values of a and bis tan 2x lim = 0? 20. a. The U.S. Postal Service will accept a box for domestic ship- ment only if the sum of its length and girth (distance around) does not exceed 108 in. What dimensions will give a box with a square end the largest possible volume? Girth = distance around here Length Square end 22. A window is in the form of a rectangle surmounted by a semicircle. The rectangle is of clear glass, whereas the semicircle is of tinted glass that transmits only half as much light per unit area as clear glass does. The total perimeter is fixed. Find the proportions of the window that will admit the most light. Neglect the thickness of the frame. 31. A wire b m long is cut into two pieces. One piece is bent into an equilateral triangle and the other is bent into a circle. If the sum of the areas enclosed by each part is a minimum, what is the length of each part? 32. Answer Exercise 31 if one piece is bent into a square and the other into a circle. 46. Two masses hanging side by side from springs have positions si = 2 sin t and s2 = sin 2t, respectively. a. At what times in the interval 0 < 1 do the masses pass each other? (Hint: sin 21 = 2 sint cost.) b. When in the interval 0 51 527 is the vertical distance between the masses the greatest? What is this distance? (Hint: cos 2t = 2 cost - 1.) Model wwwww my 0 32 m2 S 50. Airplane landing path An airplane is flying at altitude H when it begins its descent to an airport runway that is at horizontal ground distance L from the airplane, as shown in the figure. Assume that the landing path of the airplane is the graph of a cubic polynomial func- tion y = ax + bx² + cx + d, where y-L) = H and y(0) = 0. a. What is dy/dx at x = 0? b. What is dy/dx at x = -L? c. Use the values for dy/dx at x = 0 and x = -L together with y(O) = 0 and y(-1) = H to show that y(x) = H =H12(E)' + 3 (4) ] Landing path H Cruising altitude Airport L 78. fixes xe" dx = xe – e + C 82. ./6 (sin-+ x) dx = x(sin-x)2 – 2x + 2V1 - x? sin x + C ds 98. = cost + sint, S(T) = 1 dt dv 104. dt 8 1 +12 + sect, v(0) = 1 de 1 2' 8(0) = V2 110. 0; (0) = -2, 6'(0) d13 111 (4) = -sint testi 114. a. Find a curve y = f(x) with the following properties: dy i) 6x dx? ii) Its graph passes through the point (0, 1) and has a hori- zontal tangent there. b. How many curves like this are there? How do you know? 122. Stopping a motorcycle The State of Illinois Cycle Rider Safety Program requires motorcycle riders to be able to brake from 30 mph (44 ft/sec) to 0 in 45 ft. What constant decelera- tion does it take to do that?
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