Statement.1 : The function `f (x) =lim _(xtooo) (log _(e)(1+x) -x ^(2n) sin (2x))/(1+ x ^(2n))` is discontinuous at `x=1`
Statement.2: `L.H.L. =R.H.L.ne f (1).`

Options

A. a) Statement:1 is ture , Statement:2 is true and Statement:2 is correct explanation for Statement-1
B. b) Statement-1 is true, Statement-2 is true and Statement-2 is not the correct explanation for Statemetn-1
C. c) Statement-1 is true, Statement-2 is false
D. d) Statement-1 is false, Statement-2 is true

Answer

To solve the problem, we need to analyze the function given in Statement 1 and check its continuity at \( x = 1 \).

### Step-by-Step Solution:

1. **Define the Function**:
We have the function
\[ f(x) = \lim_{x \to 0} \frac{\log_e(1+x) - x^{2n} \sin(2x)}{1 + x^{2n}}. \]

2. **Evaluate the Left-Hand Limit (LHL)**:
To find the left-hand limit as \( x \) approaches 1 from the left (\( x \to 1^{-} \)):
\[ LHL = \lim_{x \to 1^{-}} \frac{\log_e(1+x) - x^{2n} \sin(2x)}{1 + x^{2n}}. \]
Substitute \( x = 1 \):
\[ LHL = \frac{\log_e(2) - 1^{2n} \sin(2)}{1 + 1^{2n}} = \frac{\log_e(2) - \sin(2)}{2}. \]

3. **Evaluate the Right-Hand Limit (RHL)**:
Now, we find the right-hand limit as \( x \) approaches 1 from the right (\( x \to 1^{+} \)):
\[ RHL = \lim_{x \to 1^{+}} \frac{\log_e(1+x) - x^{2n} \sin(2x)}{1 + x^{2n}}. \]
Substitute \( x = 1 \):
\[ RHL = \frac{\log_e(2) - 1^{2n} \sin(2)}{1 + 1^{2n}} = \frac{\log_e(2) - \sin(2)}{2}. \]

4. **Check Continuity**:
To check if \( f(x) \) is continuous at \( x = 1 \), we need to compare the limits:
\[ LHL = RHL = \frac{\log_e(2) - \sin(2)}{2}. \]
Now we need to find \( f(1) \):
\[ f(1) = \lim_{x \to 1} \frac{\log_e(1+x) - x^{2n} \sin(2x)}{1 + x^{2n}}. \]
Substitute \( x = 1 \):
\[ f(1) = \frac{\log_e(2) - \sin(2)}{2}. \]

5. **Conclusion**:
Since \( LHL = RHL = f(1) \), the function is continuous at \( x = 1 \). Therefore, Statement 1 is false.

6. **Evaluate Statement 2**:
Statement 2 claims that \( LHL \neq RHL \) and \( f(1) \). Since we found that \( LHL = RHL = f(1) \), Statement 2 is also false.

### Final Answer:
- Statement 1 is false.
- Statement 2 is false.