The integer ‘n’ for which the value of `lim _( x to oo) (( 1- cos x) (e ^(x) - cos x) - (x ^(3))/( 2))/( x ^(n))` is a finite non zero number, is____.

Explanation

To solve the limit problem, we need to analyze the expression given:

\[ \lim_{x \to \infty} \frac{(1 - \cos x)(e^x - \cos x) - \frac{x^3}{2}}{x^n} \]

### Step 1: Expand \(1 - \cos x\) and \(e^x - \cos x\)

The Taylor series expansion for \( \cos x \) around \( x = 0 \) is:

\[ \cos x \approx 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots \]

Thus,

\[ 1 - \cos x \approx \frac{x^2}{2} - \frac{x^4}{24} + \cdots \]

The Taylor series expansion for \( e^x \) is:

\[ e^x \approx 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \cdots \]

So,

\[ e^x - \cos x \approx (1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \cdots) - (1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots) \]

This simplifies to:

\[ e^x - \cos x \approx x + x^2 + \frac{x^3}{6} - \frac{x^4}{24} + \cdots \]

### Step 2: Combine the expansions

Now we can combine the two expansions:

\[ (1 - \cos x)(e^x - \cos x) \approx \left(\frac{x^2}{2}\right)\left(x + x^2 + \frac{x^3}{6} - \frac{x^4}{24}\right) \]

Calculating this product, we focus on the leading term:

\[ \approx \frac{x^2}{2} \cdot x = \frac{x^3}{2} \]

### Step 3: Substitute back into the limit expression

Now we substitute this back into our limit expression:

\[ \lim_{x \to \infty} \frac{\left(\frac{x^3}{2}\right) - \frac{x^3}{2}}{x^n} \]

This simplifies to:

\[ \lim_{x \to \infty} \frac{0}{x^n} = 0 \]

### Step 4: Consider higher-order terms

Next, we need to consider the next leading term from the product \( (1 - \cos x)(e^x - \cos x) \). The next term will be:

\[ \frac{x^2}{2} \cdot x^2 = \frac{x^4}{2} \]

Thus, we have:

\[ \lim_{x \to \infty} \frac{\frac{x^4}{2}}{x^n} \]

### Step 5: Find the value of \(n\)

For the limit to be a finite non-zero number, we need:

\[ \frac{x^4}{2x^n} = \frac{1}{2} x^{4-n} \]

This will be finite and non-zero if \(4 - n = 0\), which gives:

\[ n = 4 \]

### Conclusion

The integer \(n\) for which the limit is a finite non-zero number is:

\[ \boxed{4} \]