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 Cx 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 ))
A constant is proven using real analysis.