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MA1421-08 S3 - 1 MA1421 Basic Applied Mathematics for...

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1 MA1421 Basic Applied Mathematics for Sciences Tutorial 3 Answers 1. (a) lim x 0 x - sin x x ( 0 0 form) = lim x 0 1 - cos x 1 =0 . (b) lim x 0 x - sin x x 3 ( 0 0 form) = lim x 0 1 - cos x 3 x 2 ( 0 0 form) = lim x 0 sin x 6 x ( 0 0 form) = lim x 0 cos x 6 = 1 6 . (c) lim x →∞ x 3 e x ( form) = lim x →∞ 3 x 2 e x ( form) = lim x →∞ 6 x e x ( form) = lim x →∞ 6 e x =0 . (d) lim x 0 + x ln x = lim x 0 + ln x 1 x ( form) = lim x 0 + 1 x - 1 x 2 = lim x 0 + ( - x ) =0 .
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2 (e) lim x 0 + x x = lim x 0 + e x ln x ( e y is a continuous function of y ) = e lim x 0 + x ln x = e 0 = 1 (f) lim x 0 e x - 1 x = lim x 0 1 + x 1! + x 2 2! + ... - 1 x = lim x 0 1 + x 2! + x 2 3! + ... =1 (Note g ( x ) = x 2! + x 2 3! + ... is continuous. Here g (0) = 0) 2. If - 1 < x 8 < 1, i.e., - 8 < x < 8, we have (8 + x ) 1 2 = 8(1 + x 8 ) 1 2 8[1 + 1 2 1 x 8 · + 1 2 2 x 8 · 2 + 1 2 3 x 8 · 3 ] = 8[1 + 1 2 1! x 8 · + 1 2 ( 1 2 - 1) 2! x 8 · 2 + 1 2 ( 1 2 - 1)( 1 2 - 2) 3! x 8 · 3 ] = 8[1 + 1 16 x - 1 2 · 1 2 2 · 1 8 2 x 2 + 1 2 · 1 2 · 3 2 3 · 2 · 1 8 3 x 3 ] = 8[1 + 1 16 x - 1 8 3 x 2 + 1 2 · 8 4 x 3 ] 3. sin x = x - x 3 3! + x 5 5! - x 7 7! + ... d dx sin x = d dx x - d dx x 3 3! + d dx x 5 5! - d dx x 7 7! + ... cos x = 1 - x 2 2! + x 4 4! - x 6 6! + ...
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3 4. (1 - x + x 2 ) 1 2 =[1 + ( - x + x 2 )] 1 2 =1 + 1 2 1 ( - x + x 2 ) + 1 2 2 ( - x + x 2 ) 2 + 1 2 3 ( - x + x 2 ) 3 + ... =1 + 1 2 1 ( - x ) + o ( x ) =1 - 1 2 x + o ( x ) 5. (a) Let u = 4 - x 2 . Then du = - 2 x dx Z x (4 - x 2 ) 3 2 dx = - 1 2 ¶ Z 1 u 3 2 du = - 1 2 " u - 3 2 +1 - 3 2 + 1 # = u - 1 2 + c =(4 - x 2 ) - 1 2 + c
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