\[\large f(x+y)=\frac{f(x)+f(y)}{1-f(x)f(y)}\] The function \(f(x)\) is defined for any value of \(x\) in certain open interval centered at 0, and for any values of \(x\) and \(y \) in that interval the given functional equation is always valid. Additionally, \(\displaystyle \lim_{x\to 0} \frac{f(x)}{x}=2\).

Find the approximate value of \(f(\frac{\pi}{12})\) rounded to the nearest hundredth.

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