q2s - The last inequality follows from the inequality 1 n...

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Math 310 - Quiz 2 Solutions Monday, 28 Sept 2009 No calculators, notes or text allowed. Each problem is worth ten points. 1. Use the definition of limit to prove that: lim n →∞ n - 1 2 n + 3 = 1 2 . Let ε > 0 be given and set N := 5 4 ε . Let n N and suppose that n > N . Then, n > 5 4 ε and so 1 n < 4 ε 5 . Hence, we have ± ± ± ± n - 1 2 n + 3 - 1 2 ± ± ± ± = ± ± ± ± 2( n - 1) - 1(2 n + 3) 2(2 n + 3) ± ± ± ± = ± ± ± ± - 5 4 n + 6 ± ± ± ± (note that 4 n + 6 > 0 and | - 5 | = 5) = 5 4 n + 6 < 5 4 n < 5 4 · 4 ε 5 = ε.
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Unformatted text preview: The last inequality follows from the inequality 1 n &lt; 4 5 . Hence, n-1 2 n +3 2 5 as desired. 2 Use the denition to prove that: lim n n = + . Let M &gt; 0 be given and set N := M 2 . Let n N and suppose that n &gt; N . Then, n &gt; M 2 and so n &gt; N 2 = N. Hence, n + ....
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