Find the limit using a calculator
We have $u_0 = 6$ and $u_{n+1} = \dfrac{1}{2} u_n + \dfrac{1}{u_n}$. We can use our graphing calculator to make a 'web diagram' (no idea what it is called in English, and I can't find it. It sometimes resembles a spider's web).
When I use my calculator for very high values of n I get the same answer, $12.164$.
Is this the limit?
How would I be able to obtain this limit without the graphing calculator? Is it just the intersection with the line $y=x$?
4 Answers
$\begingroup$When $x>\sqrt{2}$, one has $\sqrt{2}<\frac{x}{2}+\frac{1}{x}<x$, then $u_0>u_1>\dots>\sqrt{2}$. So the series converges, say $u_n\to u$, then $u=\sqrt{2}$ by solving $\frac{u}{2}+\frac{1}{u}=u$.
(For completeness: if $0<u_0<\sqrt{2}$, then $u_1>\sqrt{2}$, the result will be the same; If $u_0<0$, then the limit is $-\sqrt{2}$ by symmetry.)
$\endgroup$ 1 $\begingroup$Assuming the sequence $\,\{u_n\}\,$ converges to a limit $\,u\,$ ,we get from arithmetic of limits:
$$u=\frac12u+\frac1 u\implies\frac12u=\frac1u\implies u^2=2\implies u=\sqrt 2$$
$\endgroup$ 13 $\begingroup$Just for fun, the OP was actually iterating $$u_n=\frac{1}{2} u_{n-1}+\frac{1}{u_{n-1}}+6$$ If this has a limit, it is found by $u=0.5u+\frac{1}{u}+6$ or $\frac{u}{2}-\frac{1}{u}-6=0$ or $\frac{u^2}{2}-6u-1=0$.
This has exactly one positive solution, namely $6+\sqrt{38}\approx 12.1644$.
$\endgroup$ $\begingroup$If you want to search, the common English term is Cobweb Diagram
$\endgroup$
