Question

If $\left(x\right)=\frac{1}{3}\left(f\right(x+1)+\frac{5}{f(x+2)})andf\left(x\right)>0forallxϵR,then\underset{x\to \infty }{lim}f\left(x\right)is$

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

$\underset{x\to \infty }{lim}f\left(x\right)=\underset{x\to \infty }{lim}f(x+1)=\underset{x\to \infty }{lim}f(x+2)=\lambda \left(say\right)then\lambda =\frac{1}{3}(\lambda +\frac{5}{\lambda })\Rightarrow 2\lambda =\frac{5}{\lambda }\Rightarrow \lambda =\sqrt{\frac{5}{2}}(\because f(x)>0)\therefore \underset{x\to \infty }{lim}f\left(x\right)=\sqrt{\frac{5}{2}}$