# quiz 4 - x → ∞ So by the Squeeze Theorem lim x →∞ f...

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Math 1B: Calculus February 24, 2010 Quiz 4 Lecturer: Prof. Mina Aganagic GSI: Gary Sivek Name: Answers Determine whether the following sequences converge or diverge. If they converge, ±nd the corresponding limit. (Justify your answer.) 1. (2 pts) a n = log 4 n 3 2+sin n 4 To be perfectly rigorous, we will de±ne a function f ( x )= log 4 x 3 2+sin x 4 which agrees with the sequence { a n } at positive integer values of x .T h e n i f f ( x )ha sal im i t ,the sequence will have the same limit. Now | sin x 4 |≤ 1fora l l x ,so1 2+sin x 4 3. Note that this means we will not be able to apply L’Hopital’s rule! However, we do have for x 1: log 4 x 3
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Unformatted text preview: x → ∞ . So by the Squeeze Theorem, lim x →∞ f ( x ) = ∞ , and thus lim n →∞ a n = ∞ as well. 2. (3 pts) b n = n ! 11 n There are many ways to show that this approaches in±nity. As one possibility, observe that whatever b 21 is, it is positive, and that for n ≥ 22 we have: b n b n-1 = n ! 11 n · 11 n-1 ( n-1)! = n 11 ≥ 2 So b 22 ≥ 2 b 21 , and b 23 ≥ 2 b 22 ≥ 2 2 b 21 , and b 24 ≥ 2 b 23 ≤ 2 3 b 21 , and so on. In general, for any positive n we have b 21+ n ≥ 2 n b 21 > 0, and since the middle term grows without bound, lim n →∞ b n = ∞ ....
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