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ws_3 - Mathematics for Engineers...

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Mathematics for Engineers I (Math103)... student-intranet/faculties/math/ 2009 - Fall/math103 Class Work Sheet 3 1 . 1. Evaluate each of the following limits: i*) lim x →- 2 (3 x 4 + 2 x 2 - x + 1) ii) lim x 2 2 x 2 +1 x 2 +6 x - 4 iii*) lim x →- 2 x 4 + 3 x + 6 iv) lim x 2 x 2 + x - 6 x - 2 v*) lim t →- 3 t 2 - 9 2 t 2 +7 t +3 vi) lim h 0 (2+ h ) 3 - 8 h vii*) lim t 9 9 - t 3 - t viii) lim x 9 x 2 - 81 x - 3 ix*) lim t 0 ( 1 t 1+ t - 1 t ) x) lim x 1 x - x 2 1 - x xi*) lim x 2 | x - 2 | x - 2 xii) lim x 3 2 2 x 2 - 3 x | 2 x - 3 | . 2. Use the “Sandwich Theorem ” i*) lim x 0 x 2 cos(20 πx ) = 0 ii) lim x 0 x 3 + x 2 sin( π x ) = 0 iii*) lim x →∞ sin x x iv) lim x →∞ | cos x | x v*) If 1 - x 2 6 < x sin x 2(1 - cos x ) < 1, find lim x 0 x sin x 2(1 - cos x ) . 3. Find lim x x 2 - 3 x +2 x 3 - 4 x as: (i) x 2 + (ii) x → - 2 + (iii) x 0 - (vi) x 1 + . 4*. Let f ( x ) = x if x < 0 x 2 if 0 < x 2 8 - x if x > 2 Evaluate the limit of f ( x ) as: (i) x 0 + (ii) x 0 (iii) x 1 (iv) x 2 - (v) x 2 + (vi) x 2 . 1 prepared by: Hany El-Sharkawy using “L A T E X 2 ε 1
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5. Evaluate each of the following limits: i*) lim x →∞ ( x 2 + 4 - x ) ii) lim x →-∞ sinh x iii*) lim x →∞ 3 x 2 - 6 x +4 x 3 +2 iv) lim x →∞ 7 x 2 +9 x +2 2 x 2 +1 v*) lim x →∞ 3 x 4 +2 x +1 x 2 +5 vi) lim x →∞ 2 x 4 +2 x - 1 5 x 2 +9 vii*) lim x →-∞ x +1 2 x 2 + x +1
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