If `"lt"_(x to 0)(cos^(2)x - cos x - e^(x) cos x + e^(x) -x^(2)/2)/x^(n)` is a finite non-zero number, then the integer n is:

Options

A. 2
B. 3
C. 4
D. 5

Answer

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

\[ \lim_{x \to 0} \frac{\cos^2 x - \cos x - e^x \cos x + e^x - \frac{x^2}{2}}{x^n} \]

We want to find the integer \( n \) such that this limit is a finite non-zero number.

### Step 1: Simplify the Numerator

First, we can rewrite the numerator:

\[ \cos^2 x - \cos x - e^x \cos x + e^x - \frac{x^2}{2} \]

We can group the terms involving \( \cos x \):

\[ = \cos x (\cos x - e^x) + (e^x - \frac{x^2}{2}) - \cos x \]

### Step 2: Use Taylor Series Expansion

Next, we can use the Taylor series expansions for \( \cos x \) and \( e^x \) around \( x = 0 \):

- The Taylor series for \( \cos x \) is:
\[ \cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} + O(x^6) \]

- The Taylor series for \( e^x \) is:
\[ e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + O(x^4) \]

### Step 3: Substitute the Taylor Series into the Expression

Substituting these expansions into our expression gives:

\[ \cos^2 x = (1 - \frac{x^2}{2} + \frac{x^4}{24})^2 = 1 - x^2 + \frac{x^4}{12} + O(x^6) \]

Now substituting \( \cos x \) and \( e^x \):

\[ \cos^2 x - \cos x = \left(1 - x^2 + \frac{x^4}{12}\right) - \left(1 - \frac{x^2}{2} + \frac{x^4}{24}\right) \]

This simplifies to:

\[ -\frac{x^2}{2} + \frac{x^4}{12} + \frac{x^2}{2} - \frac{x^4}{24} = \frac{x^4}{12} - \frac{x^4}{24} = \frac{x^4}{24} \]

Now for \( -e^x \cos x + e^x \):

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

This gives us a series of terms that will also contribute to the overall limit.

### Step 4: Combine and Analyze the Limit

Combining all these terms, we will find that the leading term in the numerator will be of order \( x^4 \). Thus, we can approximate:

\[ \lim_{x \to 0} \frac{\text{leading term}}{x^n} = \lim_{x \to 0} \frac{C x^4}{x^n} \]

For this limit to be finite and non-zero, we need \( n = 4 \).

### Final Answer

Thus, the integer \( n \) is:

\[ \boxed{4} \]