Question

If function \( f \) is continuous at point \( x = \pi \) and \[ f(x) = \begin{cases} kx + 1, & x \le \pi \\ \cos x, & x>\pi \end{cases} \] then the value of \( k \) is ______

• A. \( \frac{2}{\pi} \)
• B. \( -\frac{2}{\pi} \)
• C. \( \frac{1}{\pi} \)
• D. \( 0 \)

Hint:

For piecewise functions, always equate LHL and RHL at the boundary point.

Answer 1

The Correct Option is B

Concept: For continuity at \( x = a \): \[ \text{LHL} = \text{RHL} = f(a) \]
Step 1: Left-hand limit. \[ \lim_{x \to \pi^-} f(x) = k\pi + 1 \]
Step 2: Right-hand limit. \[ \lim_{x \to \pi^+} f(x) = \cos \pi = -1 \]
Step 3: Equate. \[ k\pi + 1 = -1 \] \[ k\pi = -2 \Rightarrow k = -\frac{2}{\pi} \]