`lim_(xrarrpi//2)(a^(cotx)-a^cosx)/(cotx-cosx)a gt 0` is equal to

Options

A. `log_(2)((pi)/(2))`
B. `log_e 2`
C. `log_e a`
D. `a`

Explanation

To solve the limit \( \lim_{x \to \frac{\pi}{2}} \frac{a^{\cot x} - a^{\cos x}}{\cot x - \cos x} \) where \( a > 0 \), we can follow these steps:

### Step 1: Rewrite the Limit
We start with the limit:
\[ L = \lim_{x \to \frac{\pi}{2}} \frac{a^{\cot x} - a^{\cos x}}{\cot x - \cos x} \]

### Step 2: Factor out \( a^{\cos x} \)
We can factor \( a^{\cos x} \) from the numerator:
\[ L = \lim_{x \to \frac{\pi}{2}} \frac{a^{\cos x} \left( a^{\cot x - \cos x} - 1 \right)}{\cot x - \cos x} \]

### Step 3: Analyze the Exponent
As \( x \to \frac{\pi}{2} \), both \( \cot x \) and \( \cos x \) approach 0. Thus, we can analyze the expression \( a^{\cot x - \cos x} - 1 \).

Using the limit property \( \lim_{t \to 0} \frac{a^t - 1}{t} = \log a \) (where \( a > 0 \)), we set:
\[ t = \cot x - \cos x \]
As \( x \to \frac{\pi}{2} \), \( t \to 0 \).

### Step 4: Apply the Limit Property
Now we can rewrite the limit:
\[ L = \lim_{x \to \frac{\pi}{2}} a^{\cos x} \cdot \frac{a^{\cot x - \cos x} - 1}{\cot x - \cos x} \]
By the limit property:
\[ \frac{a^{\cot x - \cos x} - 1}{\cot x - \cos x} \to \log a \text{ as } x \to \frac{\pi}{2} \]

### Step 5: Evaluate \( a^{\cos x} \)
As \( x \to \frac{\pi}{2} \), \( \cos x \to 0 \), hence:
\[ a^{\cos x} \to a^0 = 1 \]

### Step 6: Combine the Results
Putting it all together:
\[ L = 1 \cdot \log a = \log a \]

### Step 7: Conclusion
Thus, the limit evaluates to:
\[ \lim_{x \to \frac{\pi}{2}} \frac{a^{\cot x} - a^{\cos x}}{\cot x - \cos x} = \log a \]

### Final Answer
The final answer is:
\[ \log a \text{ (option 3)} \]