This exercise is used in Sections, and elsewhere. Let α > 0 and recall that (xα)′ = αxα−1 and (log x)′ = 1/x for all x > 0.
a) Prove that log x ≤ xα for x large. Prove that there exists a constant Cα such that log x ≤ Cαxα for all x ∈ [1,∞), Cα → ∞as α → 0+, and Cα → 0 as α→∞.
b) Obtain an analogue of part a) valid for ex and xα in place of log x and xα.
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