Question

If \[ f(x) = \begin{cases} \frac{1 - \sin x}{(n - 2x)^2} & \text{if} \quad x \neq \frac{\pi}{2} \log (\sin x) \cdot \log \left( 1 + \frac{\pi}{4x + x^2} \right) & \text{if} \quad x = \frac{\pi}{2} \end{cases} \] is continuous at \( x = \frac{\pi}{2} \), then \( k \) is equal to

• A. \( -\frac{1}{6} \)
• B. \( -\frac{1}{2} \)
• C. \( -\frac{1}{4} \)
• D. \( -\frac{2}{8} \)

Hint:

To ensure continuity of a piecewise function at a point, both the left-hand and right-hand limits at that point should be equal to the function value at that point.

Answer 1

The Correct Option is C

We first examine the given function and use the limit process to determine the value of \( k \).
After calculating and ensuring the continuity at \( x = \frac{\pi}{2} \), we find that the value of \( k \) is \( -\frac{1}{4} \).