# Strange Functional Equation

Algebra Level 3

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

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

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