Infinite recurrence

Calculus Level 3

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

\[\displaystyle \lim_{n\rightarrow\infty} t_n\]

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