`Lt_(x to a) sqrt((a^(2) -x^(2))) cot {pi/2sqrt((a-x)/(a+x)))}=`

Options

A. `-(4a)/pi`
B. `-(2a)/pi`
C. `(2a)/pi`
D. `(4a)/pi`

Answer

To solve the limit \( L = \lim_{x \to a} \sqrt{a^2 - x^2} \cot \left( \frac{\pi}{2} \sqrt{\frac{a - x}{a + x}} \right) \), we can follow these steps:

### Step 1: Rewrite the Limit
We start with the expression:
\[ L = \lim_{x \to a} \sqrt{a^2 - x^2} \cot \left( \frac{\pi}{2} \sqrt{\frac{a - x}{a + x}} \right) \]
As \( x \to a \), both \( \sqrt{a^2 - x^2} \) and \( \cot \left( \frac{\pi}{2} \sqrt{\frac{a - x}{a + x}} \right) \) approach 0, leading to an indeterminate form \( 0 \cdot 0 \).

### Step 2: Simplify the Square Root
We can express \( \sqrt{a^2 - x^2} \) as:
\[ \sqrt{a^2 - x^2} = \sqrt{(a - x)(a + x)} \]
Thus, we rewrite \( L \):
\[ L = \lim_{x \to a} \sqrt{(a - x)(a + x)} \cot \left( \frac{\pi}{2} \sqrt{\frac{a - x}{a + x}} \right) \]

### Step 3: Analyze the Cotangent Function
Next, we analyze the term \( \cot \left( \frac{\pi}{2} \sqrt{\frac{a - x}{a + x}} \right) \). As \( x \to a \), \( \sqrt{\frac{a - x}{a + x}} \) approaches \( 0 \):
\[ \cot \left( \frac{\pi}{2} \sqrt{\frac{a - x}{a + x}} \right) = \frac{\cos \left( \frac{\pi}{2} \sqrt{\frac{a - x}{a + x}} \right)}{\sin \left( \frac{\pi}{2} \sqrt{\frac{a - x}{a + x}} \right)} \]
Using the small angle approximation, \( \sin z \approx z \) and \( \cos z \approx 1 \) for small \( z \):
\[ \cot \left( \frac{\pi}{2} \sqrt{\frac{a - x}{a + x}} \right) \approx \frac{1}{\frac{\pi}{2} \sqrt{\frac{a - x}{a + x}}} \]

### Step 4: Substitute Back into the Limit
Now substituting back into our limit:
\[ L = \lim_{x \to a} \sqrt{(a - x)(a + x)} \cdot \frac{2}{\pi} \cdot \frac{1}{\sqrt{\frac{a - x}{a + x}}} \]
This simplifies to:
\[ L = \lim_{x \to a} \frac{2\sqrt{(a - x)(a + x)}}{\pi \sqrt{\frac{a - x}{a + x}}} \]
Rearranging gives:
\[ L = \lim_{x \to a} \frac{2\sqrt{(a + x)(a - x)}}{\pi} \cdot \sqrt{\frac{a + x}{a - x}} = \lim_{x \to a} \frac{2(a + x)}{\pi} \]

### Step 5: Evaluate the Limit
As \( x \to a \):
\[ L = \frac{2(2a)}{\pi} = \frac{4a}{\pi} \]

### Final Answer
Thus, the limit is:
\[ \boxed{\frac{4a}{\pi}} \]