`lim_(xrarr(pi^(-))/(2)) [1+(cosx)^(cosx)]^(2)=`

Options

A. Does not exist
B. `1`
C. `e`
D. `4`

Explanation

To solve the limit problem \( \lim_{x \to \frac{\pi}{2}} \left[1 + (\cos x)^{\cos x}\right]^2 \), we will follow these steps:

### Step 1: Simplify the Expression
We start with the limit:
\[ L = \lim_{x \to \frac{\pi}{2}} \left[1 + (\cos x)^{\cos x}\right]^2 \]
We need to focus on the term \( (\cos x)^{\cos x} \) as \( x \) approaches \( \frac{\pi}{2} \).

### Step 2: Analyze \( (\cos x)^{\cos x} \)
As \( x \to \frac{\pi}{2} \), \( \cos x \to 0 \). Therefore, we need to evaluate the limit of \( (\cos x)^{\cos x} \):
\[ (\cos x)^{\cos x} = e^{\cos x \cdot \log(\cos x)} \]
We will find the limit of \( \cos x \cdot \log(\cos x) \) as \( x \to \frac{\pi}{2} \).

### Step 3: Find the Limit of \( \cos x \cdot \log(\cos x) \)
We can rewrite:
\[ \lim_{x \to \frac{\pi}{2}} \cos x \cdot \log(\cos x) \]
As \( x \to \frac{\pi}{2} \), \( \cos x \to 0 \) and \( \log(\cos x) \to -\infty \). This gives us an indeterminate form of \( 0 \cdot (-\infty) \).

### Step 4: Change the Form to Apply L'Hôpital's Rule
We can rewrite the expression:
\[ \lim_{x \to \frac{\pi}{2}} \cos x \cdot \log(\cos x) = \lim_{x \to \frac{\pi}{2}} \frac{\log(\cos x)}{\frac{1}{\cos x}} \]
This is now in the form \( \frac{-\infty}{\infty} \), which allows us to apply L'Hôpital's Rule.

### Step 5: Apply L'Hôpital's Rule
Differentiate the numerator and denominator:
- The derivative of \( \log(\cos x) \) is \( -\tan x \).
- The derivative of \( \frac{1}{\cos x} \) is \( \sec x \tan x \).

Thus, we have:
\[ \lim_{x \to \frac{\pi}{2}} \frac{-\tan x}{\sec x \tan x} = \lim_{x \to \frac{\pi}{2}} -\frac{1}{\sec x} = \lim_{x \to \frac{\pi}{2}} -\cos x \]
As \( x \to \frac{\pi}{2} \), \( -\cos x \to 0 \).

### Step 6: Substitute Back
Now we have:
\[ \lim_{x \to \frac{\pi}{2}} \cos x \cdot \log(\cos x) = 0 \]
Thus,
\[ (\cos x)^{\cos x} = e^{0} = 1 \]

### Step 7: Final Limit Calculation
Now substitute back into the original limit:
\[ L = \lim_{x \to \frac{\pi}{2}} \left[1 + 1\right]^2 = \left[2\right]^2 = 4 \]

### Conclusion
The final answer is:
\[ \lim_{x \to \frac{\pi}{2}} \left[1 + (\cos x)^{\cos x}\right]^2 = 4 \]