Strange Functional Equation

Algebra Level 2

$f(x)$ 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 $\displaystyle\lim_{n\rightarrow \infty }(\ln\: a_{n})$ .