What is the value of \( a \) for which \( f(x) \) is continuous? \[ f(x) = \begin{cases} \sin x, & \text{if } x \leq \frac{\pi}{2} \\ a x, & \text{if } x > \frac{\pi}{2} \end{cases} \]
For continuity at \( x = \frac{\pi}{2} \),
LHL: \( \lim_{x \to \frac{\pi}{2}^-} f(x) = \sin \frac{\pi}{2} = 1 \)
RHL: \( \lim_{x \to \frac{\pi}{2}^+} f(x) = a \cdot \frac{\pi}{2} \)
For continuity, LHL = RHL:
\( 1 = a \cdot \frac{\pi}{2} \)
Solving for \( a \):
\( a = \frac{2}{\pi} \)
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