The integral value of `n` so that `lim_(x->0) f(x)` where `f(x) ={(sinx-x)(2sinx-ln((1+x)/(1-x)))}/x^n` is a finite non-zero number

Options

A. 2
B. 4
C. 6
D. 8

Answer

To solve the limit problem given by

\[ \lim_{x \to 0} f(x) = \frac{( \sin x - x )( 2 \sin x - \ln \left( \frac{1+x}{1-x} \right))}{x^n} \]

we need to find the integral value of \( n \) such that this limit is a finite non-zero number.

### Step 1: Expand \( \sin x \) using Taylor series
The Taylor series expansion of \( \sin x \) around \( x = 0 \) is:

\[ \sin x = x - \frac{x^3}{6} + \frac{x^5}{120} - \cdots \]

### Step 2: Substitute the expansion into \( \sin x - x \)
Using the expansion, we have:

\[ \sin x - x = \left( x - \frac{x^3}{6} + \frac{x^5}{120} - \cdots \right) - x = -\frac{x^3}{6} + \frac{x^5}{120} - \cdots \]

### Step 3: Expand \( \ln \left( \frac{1+x}{1-x} \right) \)
Using the series expansion for \( \ln(1+x) \) and \( \ln(1-x) \):

\[ \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots \]
\[ \ln(1-x) = -\left( x + \frac{x^2}{2} + \frac{x^3}{3} + \cdots \right) \]

Thus,

\[ \ln \left( \frac{1+x}{1-x} \right) = \ln(1+x) - \ln(1-x) = \left( x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots \right) - \left( -x - \frac{x^2}{2} - \frac{x^3}{3} - \cdots \right) \]

This simplifies to:

\[ \ln \left( \frac{1+x}{1-x} \right) = 2x + \frac{x^3}{3} + \cdots \]

### Step 4: Substitute back into \( f(x) \)
Now substituting back into \( f(x) \):

\[ 2 \sin x - \ln \left( \frac{1+x}{1-x} \right) = 2 \left( -\frac{x^3}{6} + \frac{x^5}{120} - \cdots \right) - \left( 2x + \frac{x^3}{3} + \cdots \right) \]

This gives:

\[ = -\frac{x^3}{3} + O(x^5) \]

### Step 5: Combine the terms
Now we have:

\[ f(x) = \frac{(-\frac{x^3}{6})(-\frac{x^3}{3})}{x^n} = \frac{\frac{x^6}{18}}{x^n} = \frac{x^{6-n}}{18} \]

### Step 6: Evaluate the limit
We need to evaluate:

\[ \lim_{x \to 0} \frac{x^{6-n}}{18} \]

For this limit to be finite and non-zero, we require \( 6 - n = 0 \), which gives \( n = 6 \).

### Conclusion
Thus, the integral value of \( n \) such that the limit is a finite non-zero number is:

\[ \boxed{6} \]