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# hw9-extra - taining a except possibly at a If f and g have...

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Homework 9: Additional Problems (required). Advanced Calculus I. Math 451, Fall 2011 1. Compute the following limits. Justify all steps. (a) lim x π 1 + x 1 - sin x (b) lim x a x n - a n x - a (c) lim x 0 1 - cos x x 2 (Hint: use the limit of sin( x ) /x .) 2. Prove the following Comparison Theorem for Functions . Let a R and let f and g be functions deﬁned on some open interval con-
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Unformatted text preview: taining a , except possibly at a . If f and g have a (real) limit as x approaches a and f ( x ) ≤ g ( x ) for all x then lim x → a f ( x ) ≤ lim x → a g ( x ) . (If needed, please only use theorems but not exercises from the textbook in your argument)....
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