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Question: If f(x) = \(\frac{1}{3}\left\{ f(x + 1) + \frac{5}{f(x + 2)} \right\}\) and f(x) \> 0 for all x Ī R,...

If f(x) = 13{f(x+1)+5f(x+2)}\frac{1}{3}\left\{ f(x + 1) + \frac{5}{f(x + 2)} \right\} and f(x) > 0 for all x Ī R, then lim⁡x→∞\lim_{x \rightarrow \infty}f(x) is

A

25\sqrt{\frac{2}{5}}

B

52\sqrt{\frac{5}{2}}

C

D

Does not exist

Answer

52\sqrt{\frac{5}{2}}

Explanation

Solution

Let lim⁡x→∞\lim_{x \rightarrow \infty}f(x) = I then,

lim⁡x→∞f(x+1)lim⁡x→∞f(x+2)=I\lim_{x \rightarrow \infty}f(x + 1)\lim_{x \rightarrow \infty}f(x + 2) = I

Since, f(x)=13{f(x+1)+5f(x+2)}f(x) = \frac{1}{3}\left\{ f(x + 1) + \frac{5}{f(x + 2)} \right\}

\ =−x(1−ax1+ax)=x(ax−1ax+1)=f(x)= - x \left( \frac { 1 - a ^ { x } } { 1 + a ^ { x } } \right) = x \left( \frac { a ^ { x } - 1 } { a ^ { x } + 1 } \right) = f ( x )

Ž I = 13(I+5I)\frac{1}{3}\left( I + \frac{5}{I} \right) Ž 3I2 = I2 + 5

Ž 2I2 = 5 Ž I =52\sqrt{\frac{5}{2}}.