If `f(x)=(3^(x)+3^(-x)-2)/(x^(2))` for `x ne 0` is continuous at x = 0, iff f(0) is equal to

Options

A. log 3
B. `(log3)^(2)`
C. `log((1)/(3))`
D. `e^(3)`

Explanation

To determine the value of \( f(0) \) such that the function \( f(x) = \frac{3^x + 3^{-x} - 2}{x^2} \) is continuous at \( x = 0 \), we need to find the limit of \( f(x) \) as \( x \) approaches 0.

### Step-by-Step Solution:

1. **Identify the Function**:
We have the function defined as:
\[ f(x) = \frac{3^x + 3^{-x} - 2}{x^2} \quad \text{for } x \neq 0 \]

2. **Evaluate the Limit as \( x \) Approaches 0**:
To find \( f(0) \), we need to calculate:
\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{3^x + 3^{-x} - 2}{x^2} \]
Substituting \( x = 0 \) directly gives us:
\[ \frac{3^0 + 3^0 - 2}{0^2} = \frac{2 - 2}{0} = \frac{0}{0} \]
This is an indeterminate form, so we apply L'Hôpital's Rule.

3. **Apply L'Hôpital's Rule**:
Differentiate the numerator and denominator:
- The derivative of the numerator \( 3^x + 3^{-x} - 2 \) is:
\[ \frac{d}{dx}(3^x) + \frac{d}{dx}(3^{-x}) = 3^x \ln(3) - 3^{-x} \ln(3) \]
- The derivative of the denominator \( x^2 \) is:
\[ \frac{d}{dx}(x^2) = 2x \]
Thus, we have:
\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{3^x \ln(3) - 3^{-x} \ln(3)}{2x} \]

4. **Evaluate the New Limit**:
Substituting \( x = 0 \) again gives us:
\[ \frac{3^0 \ln(3) - 3^0 \ln(3)}{2 \cdot 0} = \frac{0}{0} \]
We apply L'Hôpital's Rule again.

5. **Differentiate Again**:
Differentiate the numerator and denominator again:
- The derivative of the numerator \( 3^x \ln(3) - 3^{-x} \ln(3) \) is:
\[ 3^x (\ln(3))^2 + 3^{-x} (\ln(3))^2 \]
- The derivative of the denominator \( 2x \) is:
\[ 2 \]
Thus, we have:
\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{3^x (\ln(3))^2 + 3^{-x} (\ln(3))^2}{2} \]

6. **Final Evaluation**:
Now substituting \( x = 0 \):
\[ \frac{3^0 (\ln(3))^2 + 3^0 (\ln(3))^2}{2} = \frac{(\ln(3))^2 + (\ln(3))^2}{2} = \frac{2(\ln(3))^2}{2} = (\ln(3))^2 \]

Thus, we find that:
\[ f(0) = (\ln(3))^2 \]

### Conclusion:
The value of \( f(0) \) such that \( f(x) \) is continuous at \( x = 0 \) is:
\[ \boxed{(\ln(3))^2} \]