If f(x)+f(y)=f(1−xyx+y) for all x,y∈R,(xy=1),andx→0limxf(x)=2 then find f(31)andf′(1).

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{If } {f{{({x})}}}+{f{{({y})}}}={f{{({{{x}+{y}}}{{{1}-{x}{y}}}

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$If f (x) + f (y) = f (\frac{x + y}{1 - x y}) for all x, y \in R, (x y \neq = 1), and x \to 0 lim \frac{f ( x )}{x} = 2 then find f (\frac{1}{3}) and f^{'} (1) .$

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If f (x) + f (y) = f (\frac{x + y}{1 - x y}) for all x, y \in R, (x y \neq = 1), and x \to 0 lim \frac{f ( x )}{x} = 2 then find f (\frac{1}{3}) and f^{'} (1) .

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Question Text

If f (x) + f (y) = f (\frac{x + y}{1 - x y}) for all x, y \in R, (x y \neq = 1), and x \to 0 lim \frac{f ( x )}{x} = 2 then find f (\frac{1}{3}) and f^{'} (1) .