The least integer n for which `lim_(x to 0) ((cos x - 1) (cos x - e^(x)))/(x^(n))` is a finite non-zero number is

Options

A. 4
B. 3
C. 2
D. 1

Answer

To solve the limit problem, we need to find the least integer \( n \) for which

\[ \lim_{x \to 0} \frac{(\cos x - 1)(\cos x - e^x)}{x^n} \]

is a finite non-zero number. We'll approach this step by step.

### Step 1: Analyze \( \cos x - 1 \)

Using the Taylor series expansion for \( \cos x \) around \( x = 0 \):

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

Thus,

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

### Step 2: Analyze \( \cos x - e^x \)

Using the Taylor series expansion for \( e^x \) around \( x = 0 \):

\[ e^x = 1 + x + \frac{x^2}{2} + O(x^3) \]

So,

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

This simplifies to:

\[ \cos x - e^x = -x - x^2 + O(x^3) \]

### Step 3: Combine the results

Now substituting these results into our limit expression:

\[ \lim_{x \to 0} \frac{\left(-\frac{x^2}{2} + O(x^4)\right)\left(-x - x^2 + O(x^3)\right)}{x^n} \]

### Step 4: Simplify the product

The leading term in the product \( (\cos x - 1)(\cos x - e^x) \) is:

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

Thus, the limit can be rewritten as:

\[ \lim_{x \to 0} \frac{\frac{x^3}{2} + O(x^4)}{x^n} \]

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

This limit simplifies to:

\[ \lim_{x \to 0} \frac{x^3/2 + O(x^4)}{x^n} = \lim_{x \to 0} \left(\frac{1}{2} x^{3-n} + O(x^{4-n})\right) \]

For this limit to be finite and non-zero, we need the exponent of \( x \) in the leading term to be zero:

\[ 3 - n = 0 \implies n = 3 \]

### Conclusion

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

\[ \boxed{3} \]