Consider the function:
`f(x)={{:((alphacosx)/(pi-2x),"if "xne(pi)/(2)),(3,"if "x=(pi)/(2)):}`
Which is continuous at `x=(pi)/(2)`, where `alpha` is a constant.
What is `underset(xrarr0)(lim)f(x)` equal to?

Options

A. 0
B. 3
C. `(3)/(pi)`
D. `(6)/(pi)`

Answer

To solve the problem, we need to analyze the function given and find the limit as \( x \) approaches 0.

### Step-by-Step Solution:

1. **Understanding the Function**:
The function \( f(x) \) is defined as:
\[ f(x) = \begin{cases} \frac{\alpha \cos x}{\pi - 2x} & \text{if } x \neq \frac{\pi}{2} \\ 3 & \text{if } x = \frac{\pi}{2} \end{cases} \]
We need to ensure that this function is continuous at \( x = \frac{\pi}{2} \).

2. **Finding the Limit as \( x \) Approaches \( \frac{\pi}{2} \)**:
To check continuity at \( x = \frac{\pi}{2} \), we need to find:
\[ \lim_{x \to \frac{\pi}{2}} f(x) = \lim_{x \to \frac{\pi}{2}} \frac{\alpha \cos x}{\pi - 2x} \]
Substituting \( x = \frac{\pi}{2} \):
\[ f\left(\frac{\pi}{2}\right) = \frac{\alpha \cos\left(\frac{\pi}{2}\right)}{\pi - 2\left(\frac{\pi}{2}\right)} = \frac{\alpha \cdot 0}{\pi - \pi} = \frac{0}{0} \]
This is an indeterminate form, so we apply L'Hôpital's Rule.

3. **Applying L'Hôpital's Rule**:
Differentiate the numerator and denominator:
- The derivative of the numerator \( \alpha \cos x \) is \( -\alpha \sin x \).
- The derivative of the denominator \( \pi - 2x \) is \( -2 \).

Thus, we have:
\[ \lim_{x \to \frac{\pi}{2}} \frac{-\alpha \sin x}{-2} = \lim_{x \to \frac{\pi}{2}} \frac{\alpha \sin x}{2} \]
Substituting \( x = \frac{\pi}{2} \):
\[ = \frac{\alpha \cdot 1}{2} = \frac{\alpha}{2} \]

4. **Setting the Limit Equal to 3**:
Since the function is continuous at \( x = \frac{\pi}{2} \), we set:
\[ \frac{\alpha}{2} = 3 \]
Solving for \( \alpha \):
\[ \alpha = 6 \]

5. **Finding the Limit as \( x \) Approaches 0**:
Now we need to find:
\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{6 \cos x}{\pi - 2x} \]
Substituting \( x = 0 \):
\[ = \frac{6 \cdot \cos(0)}{\pi - 2 \cdot 0} = \frac{6 \cdot 1}{\pi} = \frac{6}{\pi} \]

### Final Answer:
\[ \lim_{x \to 0} f(x) = \frac{6}{\pi} \]