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Real Analysis : Proof of a Constant

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Please see the attached file for the fully formatted problem.

Let > 0. Prove that log x  x for x large. Prove that there exists a
constant C such that log x  C x for all x 2 [1, 1 ), C ! 1 as ! 0+,
and C ! 0 as ! 1 Please justify all steps and be rigorous because it is an analysis problem.
(Note: The problem falls under the chapter on Differentiability on R in
the section entitled The Mean Value Theorem, and the hint says: Find the
maximum of f(x) = log x/x forx 2 [1, 1 ))
1

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A constant is proven using real analysis.

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