9 - Math 310 - hw 9 solutions Friday, 20 Nov 2009 20.14,...

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Math 310 - hw 9 solutions 20.14, 20.20.a, 28.2; Friday, 20 Nov 2009 20.14 Prove that lim x 0 + 1 x = + and lim x 0 - 1 x = -∞ . Let ( x n ) n be a sequence in (0 , ) such that x n 0. We prove that 1 x n + . Let M > 0, then since x n 0, there is N such that for all n N , n > N implies x n = | x n - 0 | < 1 M . Hence, for n N , n > N implies 1 x n > M . Thus, 1 x n + . Therefore, by Definitions 20.1 and 20.3(b), we have lim x 0 + 1 x = + . Note that one may also use Theorem 9.10. The second assertion is proved similarly. Let ( x n ) n be a sequence in ( -∞ , 0) such that x n 0. We prove that 1 x n → -∞ . Let M < 0, then since x n 0, there is N such that for all n N , n > N implies - x n = | x n - 0 | < - 1 M . Hence, for n N , n > N implies 1 x n < M . Thus, 1 x n → -∞ . Therefore, by Definitions 20.1 and 20.3(b), we have lim x 0 - 1 x = -∞ . ± 20.20.a Let
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