Saturday, 28 September 2019

real analysis - When limxrightarrow+inftyfracf(x)x=0 implies limxrightarrow+inftyf(x)=0?

I have just solved a problem:




Let f:[0,+)R be continuous on [0,+) and differentiable on (0,+). If lim, prove that \displaystyle \lim_{x\rightarrow +\infty}\frac{f(x)}{x}=0





My questions is about the inverse: if \displaystyle \lim_{x\rightarrow +\infty}\frac{f(x)}{x}=0, which hypothesis can be added (if needed) so that we can conclude \displaystyle \lim_{x\rightarrow +\infty}f'(x)=0 ?



Actually, I tried to solve the following one, but I still cannot solve it:




If f:[0,+\infty)\rightarrow [0,+\infty) such that its second derivative is continuous, f'\le 0 and |f''|\le M for some M for all x\ge 0, then \displaystyle \lim_{x\rightarrow +\infty}f'(x)=0.




From the hypothesis, f decreases and bounded below, so f has a limit when x tends to infinity. Thus \displaystyle \lim_{x\rightarrow +\infty}\frac{f(x)}{x}=0.




My questions seems to be in a wider range than the second problem above. Any help would be appreciated.

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