If `f(x)={{:((1-tanx)/(4x-pi)",",x ne(pi)/(4)),(k ",",x=(pi)/(4)):}` is continuous at `x=(pi)/(4)` then the value of k is

Options

A. 1
B. `-1`
C. `(1)/(2)`
D. `-(1)/(2)`

Answer

To determine the value of \( k \) such that the function

\[ f(x) = \begin{cases} \frac{1 - \tan x}{4x - \pi} & \text{if } x \neq \frac{\pi}{4} \\ k & \text{if } x = \frac{\pi}{4} \end{cases} \]

is continuous at \( x = \frac{\pi}{4} \), we need to ensure that the limit of \( f(x) \) as \( x \) approaches \( \frac{\pi}{4} \) is equal to \( f\left(\frac{\pi}{4}\right) \).

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

We start by calculating the left-hand limit (LHL) and right-hand limit (RHL) of \( f(x) \) as \( x \) approaches \( \frac{\pi}{4} \):

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

### Step 2: Substitute \( x = \frac{\pi}{4} \) into the limit expression

At \( x = \frac{\pi}{4} \), we know that \( \tan\left(\frac{\pi}{4}\right) = 1 \). Therefore, we can substitute this into the limit:

\[ \lim_{x \to \frac{\pi}{4}} \frac{1 - \tan x}{4x - \pi} = \frac{1 - 1}{4\left(\frac{\pi}{4}\right) - \pi} = \frac{0}{0} \]

This is an indeterminate form, so we can apply L'Hôpital's Rule.

### Step 3: Apply L'Hôpital's Rule

We differentiate the numerator and denominator separately:

- The derivative of the numerator \( 1 - \tan x \) is \( -\sec^2 x \).
- The derivative of the denominator \( 4x - \pi \) is \( 4 \).

Thus, we have:

\[ \lim_{x \to \frac{\pi}{4}} \frac{1 - \tan x}{4x - \pi} = \lim_{x \to \frac{\pi}{4}} \frac{-\sec^2 x}{4} \]

### Step 4: Substitute \( x = \frac{\pi}{4} \) into the new limit expression

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

\[ \sec^2\left(\frac{\pi}{4}\right) = 2 \quad \text{(since } \sec\left(\frac{\pi}{4}\right) = \sqrt{2}\text{)} \]

So we get:

\[ \lim_{x \to \frac{\pi}{4}} \frac{-\sec^2 x}{4} = \frac{-2}{4} = -\frac{1}{2} \]

### Step 5: Set the limit equal to \( k \)

For \( f(x) \) to be continuous at \( x = \frac{\pi}{4} \), we need:

\[ k = \lim_{x \to \frac{\pi}{4}} f(x) = -\frac{1}{2} \]

### Conclusion

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

\[ \boxed{-\frac{1}{2}} \]