Consider the function \(f\) defined by : \(f(x) = \begin{cases} \frac{\lambda \cos(x)}{\pi/2 - x}, & x \neq \pi/2 \\ 1, & x = \pi/2 \end{cases}\) What should be the value of \(\lambda\) so that \(f\) would be continuous at \(x = \pi/2\) ?
\(1\)
Continuity at \(x = \pi/2\): \(\lim_{x \to \pi/2} f(x) = f(\pi/2) = 1\)
Put \(t = \pi/2 - x\), so \(t \to 0\) and \(\cos x = \cos(\pi/2 - t) = \sin t\)
\(\lim_{x \to \pi/2} \frac{\lambda \cos x}{\pi/2 - x} = \lambda \lim_{t \to 0} \frac{\sin t}{t} = \lambda\)
\(\lambda = 1\)
Hence, \(1\).
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