Existence of limit for sequence $x_n=\frac12\left(x_{n-1}+\frac8{x_{n-2}}\right)$ with initial values $x_0=5,x_1=10$

Let $$x_{n+1}=\tfrac{1}{2}(x_n+\frac{a}{x_{n-1}})$$

Then for $d_n=x_n-\sqrt{a}$, \begin{align} x_{n+1}-\sqrt{a}&=\tfrac{1}{2}(x_n-\sqrt{a})+\frac{a}{2}\left(\frac{1}{x_{n-1}}-\frac{1}{\sqrt{a}}\right)\\ d_{n+1}&=\tfrac{1}{2}d_n-\frac{\sqrt{a}}{2}\frac{d_{n-1}}{d_{n-1}+\sqrt{a}}=\tfrac{1}{2}d_n-\frac{1}{2}\frac{d_{n-1}}{\frac{d_{n-1}}{\sqrt{a}}+1}\\ \end{align}

So if $|d_{n-1}|<\sqrt{a}/3$, $$|d_{n+1}|\le \begin{cases}\tfrac{1}{2}|d_n|,&d_{n-1}d_n>0\\ \frac{1}{2}|d_n|+\frac{3}{4}|d_{n-1}|,&d_{n-1}d_n<0\end{cases}$$ Since the worst case cannot happen twice in succession, we must have $$|d_{n+2}|\le\tfrac{1}{4}|d_n|+\tfrac{3}{8}|d_{n-1}|$$

This recurrence inequality can be solved, $|d_n|\le A|r_1|^n+B|r_2|^n+C|r_3|^n\to0$ since $r_1\approx0.84$, $|r_2|=|r_3|\approx0.67$.

Hence, as long as some $d_k$ comes close enough to $\sqrt{a}$, $x_n\to\sqrt{a}$. (In fact, the sequence may converge to $-\sqrt{a}$, e.g. $x_0=x_1=-1$ for $a=8$. )