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Unformatted text preview: Math 1A Quiz 2 Solutions February 8th, 2008 1. (Version 1:) Calculate: lim x →∞ x 4 + x 3 + 2 + sin x √ 1 + 2 x 8 + 2 x 4 + 3 (Version 2:) Calculate: lim x →∞ 3 x 4 + x 3 + 2 √ 1 + 2 x 8 + 2 x 4 + 3 + sin x Solution: Both versions are solved by the same method: divide both the numerator and denominator by x 4 . Here’s how version 1 ends up: lim x →∞ 1 + 1 x + 2 x 4 + sin x x 4 1 x 4 √ 1 + 2 x 8 + 2 + 3 x 4 . Now, most of these terms go to 0 as x → ∞ since they’re just constants over some power of x . Nothing changes about the 1 in the numerator and the 2 in the denominator. The only terms we need to deal with are the sin x x 4 term and the term involving the square root. Now, 1 ≤ sin x ≤ 1, so also 1 x 4 ≤ sin x x 4 ≤ 1 x 4 . As x → ∞ , both 1 x 4 and 1 x 4 go to 0. The Squeeze theorem tells us that also sin x x 4 goes to 0. We just need to deal now with the square root term. We have that 1 x 4 = 1 x 4 = q 1 x 8 . Hence 1 x 4 √ 1 + 2 x 8 =...
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This note was uploaded on 09/17/2011 for the course MATH 1A taught by Professor Wilkening during the Spring '08 term at Berkeley.
 Spring '08
 WILKENING
 Math, Calculus

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