These functions drive me crazy(Part II)!

Calculus Level 4

If f(x) be a function satisfying the condition

$f(x)=\frac{1}{3}[f(x+6)+\frac{6}{f(x+7)}]$

and f(x) is greater than zero for every x belongs to R(set of real no) then

$lim_{x\rightarrow\infty} f(x)=\sqrt{m}$

Find the value of m

6 1 2 3 4 5 0

\infty

1 solution

Chew-Seong Cheong
Jul 24, 2015

$\begin{aligned} f(x) & = \frac{1}{3}\left(f(x+6) + \frac{6}{f(x+7)} \right) = \frac{f(x+6)}{3} + \frac{2}{f(x+7)} \\ \lim_{x \to \infty} f(x) & = \lim_{x \to \infty} \left(\frac{f(x+6)}{3} + \frac{2}{f(x+7)} \right) \\ \sqrt{m} & = \frac{\sqrt{m}}{3} + \frac{2}{\sqrt{m}} \\ \Rightarrow m & = \frac{m}{3} + 2 \\ \frac{2m}{3} & = 2 \\ m & = \boxed{3} \end{aligned}$

These functions drive me crazy(Part II)!

1 solution

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