If `f(n+2)=(1)/(2){f(n+1)+(9)/(f(n))}, n in N and f(n) gt0` for all ` n in N`, then ` lim_( n to oo)f(n)` is equal to

Options

A. 3
B. `-3`
C. `(1)/(2)`
D. none of these

Answer

To solve the problem, we need to analyze the given recurrence relation:

\[ f(n+2) = \frac{1}{2} \left( f(n+1) + \frac{9}{f(n)} \right) \]

We are tasked with finding:

\[ \lim_{n \to \infty} f(n) \]

### Step 1: Assume the Limit Exists
Let \( L = \lim_{n \to \infty} f(n) \). Since \( f(n) > 0 \) for all \( n \in \mathbb{N} \), we can assume \( L > 0 \).

### Step 2: Substitute the Limit into the Recurrence Relation
As \( n \) approaches infinity, we can substitute \( f(n) \), \( f(n+1) \), and \( f(n+2) \) with \( L \):

\[ L = \frac{1}{2} \left( L + \frac{9}{L} \right) \]

### Step 3: Simplify the Equation
Multiply both sides by 2 to eliminate the fraction:

\[ 2L = L + \frac{9}{L} \]

### Step 4: Rearrange the Equation
Rearranging gives:

\[ 2L - L = \frac{9}{L} \]

This simplifies to:

\[ L = \frac{9}{L} \]

### Step 5: Multiply Both Sides by \( L \)
To eliminate the fraction, multiply both sides by \( L \):

\[ L^2 = 9 \]

### Step 6: Solve for \( L \)
Taking the square root of both sides gives:

\[ L = 3 \quad \text{or} \quad L = -3 \]

### Step 7: Consider the Positive Solution
Since \( f(n) > 0 \) for all \( n \), we discard \( L = -3 \) and accept:

\[ L = 3 \]

### Conclusion
Thus, we conclude that:

\[ \lim_{n \to \infty} f(n) = 3 \]