Saturday, 3 September 2016

Limit limxtoinftyxtan1(f(x)/(x+g(x)))

I am investigating the limit




lim



given that f(x)\to0 and g(x)\to0 as x\to\infty. My initial guess is the limit exists since the decline rate of \tan^{-1} will compensate the linearly increasing x. But I'm not sure if the limit can be non zero. My second guess is the limit will always zero but I can't prove it. Thank you.



EDIT 1: this problem ca be reduced into proving that \lim_{x\to\infty}x\tan^{-1}(M/x)=M for any M\in\mathbb{R}. Which I cannot prove it yet.



EDIT 2: indeed \lim_{x\to\infty}x\tan^{-1}(M/x)=M for any M\in\mathbb{R}. Observe that



\lim_{x\to\infty}x\tan^{-1}(M/x)=\lim_{x\to0}\frac{\tan^{-1}(Mx)}{x}. By using L'Hopital's rule, the right hand side gives M. So the limit which is being investigated is equal to zero for any f(x) and g(x) as long as f(x)\to0 and g(x)\to0 as x\to\infty. The problem is solved.

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