Limit $lim_{xtoinfty}xtan^{-1}(f(x)/(x+g(x)))$

I am investigating the limit

$$\lim_{x\to\infty}x\tan^{-1}\left(\frac{f(x)}{x+g(x)}\right)$$

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.