If f(x) is a continuous function satisfying f(x)f(1/x) =f(x)+f(1/x) and f(1) `gt` 0 then `lim_(x to 1)` f(x) is equal to

Options

A. 2
B. 1
C. 3
D. none of these

Answer

To solve the problem step by step, we start by analyzing the given function and its properties.

### Step 1: Understand the given functional equation
We are given that \( f(x)f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right) \).

### Step 2: Substitute \( x = 1 \)
To find the value of \( f(1) \), we can substitute \( x = 1 \) into the functional equation:
\[ f(1)f(1) = f(1) + f(1) \]
This simplifies to:
\[ f(1)^2 = 2f(1) \]

### Step 3: Rearranging the equation
Rearranging the equation gives us:
\[ f(1)^2 - 2f(1) = 0 \]
Factoring out \( f(1) \):
\[ f(1)(f(1) - 2) = 0 \]

### Step 4: Solving for \( f(1) \)
This gives us two possible solutions:
1. \( f(1) = 0 \)
2. \( f(1) = 2 \)

### Step 5: Using the condition \( f(1) > 0 \)
Since we are given that \( f(1) > 0 \), we can discard the solution \( f(1) = 0 \). Therefore, we have:
\[ f(1) = 2 \]

### Step 6: Finding the limit as \( x \) approaches 1
Since \( f(x) \) is continuous at \( x = 1 \), we have:
\[ \lim_{x \to 1} f(x) = f(1) \]
Thus:
\[ \lim_{x \to 1} f(x) = 2 \]

### Conclusion
The limit as \( x \) approaches 1 of \( f(x) \) is:
\[ \lim_{x \to 1} f(x) = 2 \]