Monday, 12 February 2018

calculus - f is an unbounded monotonic increasing function: limlimitsxtoinftyfrac1xintx0f(t)dt=infty



This is not homework.



I'd love your help proving that if f is an unbounded monotonic increasing function, then lim I want to use it in a couple of proofs, but I can't prove it by myself.



Thanks a lot.



Answer



I think the l'Hôpital's rule may be helpful here if your f satisfies the requirements of the rule. And say the antiderviative of f is F(x), then



\lim_{x\to\infty}\frac{1}{x}\int_{0}^{x}f(t)dt = \lim_{x\to\infty}\frac{F(x) - F(0)}{x} = \lim_{x\to\infty}f(x) .



Since your tag is calculus, not something like real analysis, I assume your function is a relatively normal function involving Riemann integral.


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