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Unformatted text preview: 100000000 . On the other hand, the terms of this sequence tend to 2 by denition, which is not rational. 8.2. (a) lim n n n 2 +1 = 0. Let &gt; 0 and choose N = 1 . Then, n &gt; N implies n &gt; 1 , and 1 n &gt; 0 for all n N , so n + 1 n &gt; n &gt; 1 . This is n 2 + 1 n &gt; 1 so  n n 2 + 1  = n n 2 + 1 &lt; . (c) lim n 4 n +3 7 n5 = 4 7 . Let &gt; 0 and choose N = 1 49 ( 41 + 35). Then, n &gt; N implies n &gt; 1 49 ( 41 + 35) so (49 n35) &gt; 41 and &gt; 41 (49 n35) = (28 n + 21)(28 n20) 7(7 n5) = 7(4 n + 3)4(7 n5) 7(7 n5) 1 2 so  4 n + 3 7 n54 7  = 7(4 n + 3)4(7 n5) 7(7 n5) &lt; . (e) lim n 1 n sin n = 0. Let &gt; 0 and choose N = 1 . Then, n &gt; N implies n &gt; 1 and so we have 1 n &lt; . Hence,  1 n sin n  = 1 n  sin n  1 n &lt; because  sin n  1 for all n N ....
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 Spring '09

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