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# sols2 - SOLUTIONS TO HOMEWORK ASSIGNMENT#2 1 Compute the...

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SOLUTIONS TO HOMEWORK ASSIGNMENT #2 1. Compute the following limits: (a) lim x 0 sin( - x ) sin 3 x (b) lim θ 0 θ 3 (sin θ ) 2 (c) lim x 0 1 + 2 x - 1 - 2 x x (d) lim t 0 t t + sin t (e) lim z →∞ z 2 + 1 2 z 2 - 1 (f) lim x →-∞ cos x x 2 + 1 Solutions: Theses questions use the limit laws (study pages 67-70 and 76) and various algebraic steps. (a) lim x 0 sin( - x ) sin 3 x = - 1 3 lim x 0 sin x x 3 x sin 3 x = - 1 3 lim x 0 sin x x × lim x 0 3 x sin 3 x = - 1 3 (b) lim θ 0 ( θ ) 3 (sin θ ) 2 = lim θ 0 θ × lim θ 0 ( θ ) 2 (sin θ ) 2 = 0 × 1 2 = 0 (c) lim x 0 1 + 2 x - 1 - 2 x x = lim x 0 1 + 2 x - 1 - 2 x x × 1 + 2 x + 1 - 2 x 1 + 2 x + 1 - 2 x = lim x 0 1 + 2 x - (1 - 2 x ) x ( 1 + 2 x + 1 - 2 x ) = lim x 0 2 x x ( 1 + x + 1 - x ) = lim x 0 4 1 + 2 x + 1 - 2 x = 2 by setting x = 0 . The last step uses continuity of the function 4 1 + 2 x + 1 - 2 x at x = 0. (d) lim t 0 t t + sin t = lim t 0 1 1 + sin t/t = 1 lim t 0 (1 + sin t/t ) = 1 2 (e) lim z →∞ z 2 + 1 2 z 2 - 1 = lim z →∞ 1 + 1 /z 2 2 - 1 /z 2 = 2 since lim z 0 1 /z 2 = 0 (f) lim x →-∞

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sols2 - SOLUTIONS TO HOMEWORK ASSIGNMENT#2 1 Compute the...

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