If `f(x)={((1-sin((3x)/2))/(pi-3x) , x != pi/2),(lambda , x=pi/2))` be continuous at `x=pi/3` , then value of `lamda` is (A) `-1` (B) `1` (C) `0` (D) `2`

Options

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

Explanation

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

\[ f(x) = \begin{cases} \frac{1 - \sin\left(\frac{3x}{2}\right)}{\pi - 3x} & \text{if } x \neq \frac{\pi}{2} \\ \lambda & \text{if } x = \frac{\pi}{2} \end{cases} \]

is continuous at \( x = \frac{\pi}{3} \), we need to ensure that the left-hand limit (LHL) and right-hand limit (RHL) at \( x = \frac{\pi}{3} \) are equal to the function value at that point.

### Step 1: Calculate the Left-Hand Limit (LHL) as \( x \to \frac{\pi}{3} \)

We start by calculating the limit:

\[ \lim_{x \to \frac{\pi}{3}^-} f(x) = \lim_{x \to \frac{\pi}{3}^-} \frac{1 - \sin\left(\frac{3x}{2}\right)}{\pi - 3x} \]

Substituting \( x = \frac{\pi}{3} \):

\[ = \frac{1 - \sin\left(\frac{3 \cdot \frac{\pi}{3}}{2}\right)}{\pi - 3 \cdot \frac{\pi}{3}} = \frac{1 - \sin\left(\frac{\pi}{2}\right)}{\pi - \pi} = \frac{1 - 1}{0} = \frac{0}{0} \]

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

### Step 2: Apply L'Hôpital's Rule

Differentiate the numerator and denominator:

- The derivative of the numerator \( 1 - \sin\left(\frac{3x}{2}\right) \) is \( -\frac{3}{2} \cos\left(\frac{3x}{2}\right) \).
- The derivative of the denominator \( \pi - 3x \) is \( -3 \).

Now we can rewrite the limit:

\[ \lim_{x \to \frac{\pi}{3}^-} \frac{-\frac{3}{2} \cos\left(\frac{3x}{2}\right)}{-3} = \lim_{x \to \frac{\pi}{3}^-} \frac{\frac{3}{2} \cos\left(\frac{3x}{2}\right)}{3} = \frac{1}{2} \cos\left(\frac{3 \cdot \frac{\pi}{3}}{2}\right) = \frac{1}{2} \cos\left(\frac{\pi}{2}\right) = \frac{1}{2} \cdot 0 = 0 \]

### Step 3: Calculate the Right-Hand Limit (RHL) as \( x \to \frac{\pi}{3} \)

Since the function is defined the same way for both sides around \( x = \frac{\pi}{3} \), we have:

\[ \lim_{x \to \frac{\pi}{3}^+} f(x) = \lim_{x \to \frac{\pi}{3}^-} f(x) = 0 \]

### Step 4: Set the Function Value Equal to the Limits

For the function to be continuous at \( x = \frac{\pi}{3} \):

\[ f\left(\frac{\pi}{3}\right) = \lambda \]

Since both limits equal \( 0 \):

\[ \lambda = 0 \]

### Conclusion

Thus, the value of \( \lambda \) that makes the function continuous at \( x = \frac{\pi}{3} \) is:

\[ \boxed{0} \]