If $\lim _{n \rightarrow \infty} x_n$ exists and is finite, $x_1=2, x_{n+1}=\frac{a+b x_n}{b+c x_n} \forall…
If $\lim _{n \rightarrow \infty} x_n$ exists and is finite, $x_1=2, x_{n+1}=\frac{a+b x_n}{b+c x_n} \forall n \in N$ and $\mathrm{c}>\mathrm{b}>\mathrm{a}>\mathrm{o}$ then $\lim _{n \rightarrow \infty} x_n=$
$\sqrt{\frac{a b}{c}}$
$\sqrt{\frac{a}{c}}$
$\sqrt{\frac{a}{b}}$
$\sqrt{a / b}$
Solution
$\therefore \lim _{n \rightarrow \infty} x n$ enirts and fiwite
let $\lim _{n \rightarrow \infty} x n=1$
$\Rightarrow \mathrm{l}=\frac{\mathrm{a}+\mathrm{bl}}{\mathrm{b}+\mathrm{cl}} \Rightarrow \mathrm{bl}+\mathrm{cl}^2=\mathrm{a}+\mathrm{bl}$
$\begin{aligned} & \Rightarrow \mathrm{cl}^2=9 \\ & \Rightarrow \mathrm{l}^2=\frac{\mathrm{a}}{\mathrm{c}}\end{aligned}$
$\therefore \quad \Rightarrow 1=\sqrt{\frac{\mathrm{a}}{\mathrm{c}}}$