Question

If \(\lim_{n \to \infty} x_n\) exists and is finite, \(x_1 = 2\), \(x_{n+1} = \frac{a + b x_n}{b + c x_n}\) \(\forall n \in \mathbb{N}\) and \(c \gt b \gt a \gt 0\), then \(\lim_{n \to \infty} x_n =\)

• A. \(\sqrt{\frac{ab}{c}}\)
• B. \(\sqrt{\frac{a}{c}}\)
• C. \(\sqrt{\frac{c}{b}}\)
• D. \(\sqrt{\frac{a}{b}}\)

Hint:

For linear recurrence \(x_{n+1} = \frac{a + bx_n}{b + c x_n}\), set limit \(L\) and solve \(L = \frac{a + bL}{b + cL}\) to find \(\lim x_n\).

Answer 1

The Correct Option is B

Step 1: Set up the fixed-point equation.
If the limit \(L=\lim_{n\to\infty}x_n\) exists, then taking the limit of both sides of the recurrence gives
\[L = \frac{a+bL}{b+cL}\]

Step 2: Clear the fraction and simplify.
\[L(b+cL) = a+bL \implies bL + cL^2 = a+bL \implies cL^2 = a \implies L^2 = \frac{a}{c}\]

Step 3: Pick the correct sign.
Starting from \(x_1=2>0\) and with \(a,b,c\) all positive, every term \(x_{n+1}=\frac{a+bx_n}{b+cx_n}\) stays positive, since both numerator and denominator are positive whenever \(x_n\) is. So the limit L must be nonnegative, giving
\[L = \sqrt{\frac{a}{c}}\]

Step 4: Conclusion.
\[\boxed{\sqrt{\frac{a}{c}}}\]