If f(x) = `((3^(x)-1)^(2))/(sinx. "log"_(e) (1+x)) , x ne 0` is continuous at x =0 then f(0) is

Options

A. `"log"_(e) 3`
B. `2 "log"_(e)3`
C. `("log"_(e)3)^(2)`
D. None

Answer

To determine the value of \( f(0) \) for the function

\[ f(x) = \frac{(3^x - 1)^2}{\sin x \cdot \log_e(1 + x)}, \quad x \neq 0 \]

and ensure that it 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 Limit**:
We need to evaluate the limit:
\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{(3^x - 1)^2}{\sin x \cdot \log_e(1 + x)} \]

2. **Use Taylor Series Expansions**:
- For small \( x \), we can use the Taylor series expansion:
- \( 3^x \approx 1 + x \log(3) \) (since \( 3^x = e^{x \log(3)} \))
- \( \sin x \approx x \)
- \( \log_e(1 + x) \approx x \)

Thus, we can approximate:
\[ 3^x - 1 \approx x \log(3) \]

3. **Substituting the Approximations**:
Substitute these approximations into the limit:
\[ \lim_{x \to 0} \frac{(x \log(3))^2}{x \cdot x} = \lim_{x \to 0} \frac{x^2 (\log(3))^2}{x^2} \]

4. **Simplifying the Limit**:
The \( x^2 \) terms cancel out:
\[ \lim_{x \to 0} (\log(3))^2 = (\log(3))^2 \]

5. **Conclusion**:
Since the limit exists and is equal to \( (\log(3))^2 \), we can define \( f(0) \) to make the function continuous:
\[ f(0) = (\log(3))^2 \]

### Final Answer:
\[ f(0) = (\log_e(3))^2 \]