Infinite recurrence

A sequence is defined by the recurrence relation

$t_1 = 1 \text{ and } t_n = \dfrac{1}{2} \left(t_{n-1} + \dfrac{x}{t_{n-1}} \right).$

where $x$ is a positive real. Find

$\displaystyle \lim_{n\rightarrow\infty} t_n$