If `f(x)={{:((1-sqrt2sinx)/(pi-4x)",",ifxne(pi)/(4)),(a",",if x=(pi)/(4)):}` in continuous at `(pi)/(4)`, then a is equal to :

Options

A. 2
B. 2
C. 3
D. `1/4`

Explanation

To find the value of \( a \) such that the function

\[ f(x) = \begin{cases} \frac{1 - \sqrt{2} \sin x}{\pi - 4x} & \text{if } x \neq \frac{\pi}{4} \\ a & \text{if } x = \frac{\pi}{4} \end{cases} \]

is continuous at \( x = \frac{\pi}{4} \), we need to ensure that

\[ \lim_{x \to \frac{\pi}{4}} f(x) = f\left(\frac{\pi}{4}\right) = a. \]

### Step 1: Calculate the limit as \( x \) approaches \( \frac{\pi}{4} \)

First, we need to evaluate the limit:

\[ \lim_{x \to \frac{\pi}{4}} \frac{1 - \sqrt{2} \sin x}{\pi - 4x}. \]

### Step 2: Substitute \( x = \frac{\pi}{4} \)

Substituting \( x = \frac{\pi}{4} \):

- The numerator becomes \( 1 - \sqrt{2} \sin\left(\frac{\pi}{4}\right) = 1 - \sqrt{2} \cdot \frac{1}{\sqrt{2}} = 1 - 1 = 0 \).
- The denominator becomes \( \pi - 4\left(\frac{\pi}{4}\right) = \pi - \pi = 0 \).

Thus, we have an indeterminate form \( \frac{0}{0} \).

### Step 3: Apply L'Hôpital's Rule

Since we have \( \frac{0}{0} \), we can apply L'Hôpital's Rule, which states that:

\[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \]

if the limit on the right exists.

#### Differentiate the numerator and denominator:

- The derivative of the numerator \( 1 - \sqrt{2} \sin x \) is \( -\sqrt{2} \cos x \).
- The derivative of the denominator \( \pi - 4x \) is \( -4 \).

### Step 4: Rewrite the limit

Now we rewrite the limit using L'Hôpital's Rule:

\[ \lim_{x \to \frac{\pi}{4}} \frac{-\sqrt{2} \cos x}{-4} = \frac{\sqrt{2} \cos x}{4}. \]

### Step 5: Substitute \( x = \frac{\pi}{4} \) again

Now substituting \( x = \frac{\pi}{4} \):

\[ \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}. \]

Thus, we have:

\[ \lim_{x \to \frac{\pi}{4}} f(x) = \frac{\sqrt{2} \cdot \frac{1}{\sqrt{2}}}{4} = \frac{1}{4}. \]

### Step 6: Set the limit equal to \( a \)

Since \( f(x) \) is continuous at \( x = \frac{\pi}{4} \), we set:

\[ a = \lim_{x \to \frac{\pi}{4}} f(x) = \frac{1}{4}. \]

### Conclusion

Thus, the value of \( a \) is:

\[ \boxed{\frac{1}{4}}. \]