Strange Functional Equation

Algebra Level 2

f(x) f(x) is an even function with codomain R.\mathbb{R}. Its graph is symmetric about the line x=1 x=1 , and f(x1+x2)=f(x1)f(x2)f(x_{1}+x_{2})=f(x_{1})\cdot f(x_{2}) for any x1,x2[0,12] x_{1},x_{2}\in [0,\frac{1}{2}] , and f(1)>0. f(1)> 0.

Let an=f(2n+12n) a_{n}=f(2n+\frac{1}{2n}) . Find the value of limn(lnan) \displaystyle\lim_{n\rightarrow \infty }(\ln\: a_{n}) .

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