The value of `lim_(xtoa) sqrt(a^(2)-x^(2))"cot"(pi)/(2)sqrt((a-x)/(a+x))` is

Options

A. `(5050)/(pie)`
B. `100)/(pie)`
C. `-(5050)/(pie)`
D. `-(4950)/(pie)`

Explanation

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

### Step 1: Substitute \( x = a \)
First, we substitute \( x \) with \( a \) in the limit:
\[ \sqrt{a^2 - a^2} \cot \left( \frac{\pi}{2} \sqrt{\frac{a - a}{a + a}} \right) = \sqrt{0} \cot(0) \]
This results in \( 0 \cdot \infty \), which is an indeterminate form.

**Hint:** When substituting directly leads to an indeterminate form, consider rearranging the expression.

### Step 2: Rewrite the cotangent
We can rewrite the cotangent function:
\[ \cot \left( \frac{\pi}{2} \sqrt{\frac{a - x}{a + x}} \right) = \frac{1}{\tan \left( \frac{\pi}{2} \sqrt{\frac{a - x}{a + x}} \right)} \]
Thus, the limit becomes:
\[ \lim_{x \to a} \frac{\sqrt{a^2 - x^2}}{\tan \left( \frac{\pi}{2} \sqrt{\frac{a - x}{a + x}} \right)} \]

**Hint:** Converting cotangent to tangent can help simplify the limit.

### Step 3: Factor the square root
We can factor \( a^2 - x^2 \) as:
\[ \sqrt{a^2 - x^2} = \sqrt{(a - x)(a + x)} \]
So the limit now looks like:
\[ \lim_{x \to a} \frac{\sqrt{(a - x)(a + x)}}{\tan \left( \frac{\pi}{2} \sqrt{\frac{a - x}{a + x}} \right)} \]

**Hint:** Factoring can help separate the terms and make the limit easier to evaluate.

### Step 4: Analyze the limit of the tangent
As \( x \to a \), \( a - x \to 0 \) and \( a + x \to 2a \). Thus:
\[ \sqrt{\frac{a - x}{a + x}} \to \sqrt{\frac{0}{2a}} = 0 \]
This means:
\[ \tan \left( \frac{\pi}{2} \sqrt{\frac{a - x}{a + x}} \right) \to \tan(0) = 0 \]

**Hint:** Recognizing the behavior of the functions as they approach the limit can simplify the evaluation.

### Step 5: Use the limit property of tangent
We know that:
\[ \lim_{u \to 0} \frac{\tan(u)}{u} = 1 \]
Thus, we can rewrite the limit:
\[ \lim_{x \to a} \frac{\sqrt{(a - x)(a + x)}}{\frac{\pi}{2} \sqrt{\frac{a - x}{a + x}}} \cdot \frac{\sqrt{\frac{a - x}{a + x}}}{\tan \left( \frac{\pi}{2} \sqrt{\frac{a - x}{a + x}} \right)} \]

**Hint:** Using known limits can help to evaluate the limit more easily.

### Step 6: Evaluate the limit
Now we can evaluate:
\[ \lim_{x \to a} \frac{\sqrt{(a - x)(a + x)}}{\frac{\pi}{2} \sqrt{\frac{a - x}{a + x}}} = \lim_{x \to a} \frac{\sqrt{(a - x)(2a)}}{\frac{\pi}{2} \sqrt{\frac{a - x}{2a}}} \]
As \( x \to a \), this simplifies to:
\[ \frac{\sqrt{0 \cdot 2a}}{\frac{\pi}{2} \cdot 0} = \frac{0}{0} \]
This indicates we need to apply L'Hôpital's Rule or further simplify.

### Final Result
After applying L'Hôpital's Rule or further simplification, we find:
\[ \lim_{x \to a} \frac{4a}{\pi} = \frac{4a}{\pi} \]

Thus, the final answer is:
\[ \frac{4a}{\pi} \]