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m21am1sol

m21am1sol - MAT 21A SECTION D CALCULUS FIRST...

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MAT 21A SECTION D: CALCULUS FIRST MIDTERM–SOLUTIONS DATE AND TIME: FRIDAY, OCTOBER 20, 2006. 2:10–3:00. ROOM: 100 HUNT HALL INSTRUCTOR: M. MULASE Problem 1 (10 points) . This problem concerns various limits. Give your answer in its simplest form in the space provided. lim x 1 x x = 1; lim x 0 + x x = lim x 0 + 1 x = + ; lim x + x x = lim x + 1 x = 0 lim x 1 1 - x 1 - x = lim x 1 1 1 + x = 1 2 lim x 0 + 1 - x 1 - x = lim x 0 + 1 - 0 1 - 0 = 1 lim x 1 x 2 - 2 x + 1 x 2 - 1 = lim x 1 x - 1 x + 1 = 0 lim x →-∞ x 2 - 2 x + 1 x 2 - 1 = lim x →-∞ x - 1 x + 1 = lim x →-∞ x x + 1 + lim x →-∞ - 1 x + 1 = 1 lim x 1 x - 1 x 2 - 1 = lim x 1 1 x + 1 = 1 2 lim x 1 - x + 1 x 2 - 1 = lim x 1 - 1 x - 1 = -∞ lim h 0 ( x + h ) 3 - x 3 h = lim h 0 x 3 + 3 hx 2 + 3 h 2 x + h 3 - x 3 h = lim h 0 (3 x 2 + 3 hx 2 + h 2 ) = 3 x 2 1

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2 TOTAL POINTS FROM THIS PAGE = Problem 2 (10 points) . Let us consider the function f ( x ) = sin x x . (1) Find the following limits (4 points): lim x + sin x x = 0 lim x →-∞ sin x x = 0 lim x 0 sin x x = 1 lim x π/ 2 sin x x = sin(
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m21am1sol - MAT 21A SECTION D CALCULUS FIRST...

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