If `f(x) = (log_(e)(1+x^(2)tanx))/(sinx^(3)), x != 0` is continuous at x = 0 then f(0) must be defined as

Options

A. 1
B. 0
C. `1/2`
D. `-1`

Answer

To determine the value of \( f(0) \) such that the function \( f(x) = \frac{\log_e(1 + x^2 \tan x)}{\sin^3 x} \) is continuous at \( x = 0 \), we need to evaluate the limit of \( f(x) \) as \( x \) approaches 0.

### Step-by-Step Solution:

1. **Identify the limit**: We need to find \( \lim_{x \to 0} f(x) \).
\[ f(x) = \frac{\log_e(1 + x^2 \tan x)}{\sin^3 x} \]

2. **Substitute \( x = 0 \)**: Direct substitution gives us an indeterminate form \( \frac{0}{0} \) because:
- \( \tan(0) = 0 \) implies \( \log_e(1 + 0) = 0 \)
- \( \sin(0) = 0 \) implies \( \sin^3(0) = 0 \)

3. **Apply L'Hôpital's Rule**: Since we have an indeterminate form \( \frac{0}{0} \), we can apply L'Hôpital's Rule, which states that:
\[ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \]
if the limit on the right exists.

4. **Differentiate the numerator and denominator**:
- **Numerator**: Differentiate \( \log_e(1 + x^2 \tan x) \):
\[ \frac{d}{dx} \log_e(1 + x^2 \tan x) = \frac{1}{1 + x^2 \tan x} \cdot \left( 2x \tan x + x^2 \sec^2 x \right) \]
- **Denominator**: Differentiate \( \sin^3 x \):
\[ \frac{d}{dx} \sin^3 x = 3 \sin^2 x \cos x \]

5. **Set up the limit again**:
\[ \lim_{x \to 0} \frac{\frac{1}{1 + x^2 \tan x} \cdot (2x \tan x + x^2 \sec^2 x)}{3 \sin^2 x \cos x} \]

6. **Evaluate the limit**: As \( x \to 0 \):
- \( \tan x \approx x \) and \( \sec^2 x \approx 1 \)
- Therefore, \( 2x \tan x \approx 2x^2 \) and \( x^2 \sec^2 x \approx x^2 \)
- The numerator approaches \( \frac{1}{1 + 0} \cdot (2x^2 + x^2) = 3x^2 \)
- The denominator approaches \( 3 \cdot 0^2 \cdot 1 = 0 \)

Thus, we have:
\[ \lim_{x \to 0} \frac{3x^2}{3 \sin^2 x \cos x} \]

7. **Simplify further**: Since \( \sin x \approx x \) as \( x \to 0 \):
\[ \sin^2 x \approx x^2 \]
Therefore, the limit simplifies to:
\[ \lim_{x \to 0} \frac{3x^2}{3x^2 \cdot 1} = 1 \]

8. **Conclusion**: For \( f(x) \) to be continuous at \( x = 0 \), we define \( f(0) \) as:
\[ f(0) = 1 \]

### Final Answer:
Thus, \( f(0) \) must be defined as \( 1 \).

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