Question

Find the value of $k$ if
$f\left(x\right)=\{\begin{array}{c}\frac{k\cos x}{\pi -2x},x\ne \frac{\pi }{2}\\ 3,x=\frac{\pi }{2}\end{array}$ is continuous at $x=\frac{\pi }{2}$

Answer 1

$f\left(x\right)=\{\begin{array}{c}\frac{k\cos x}{\pi -2x},x\ne \frac{\pi }{2}\\ 3,x=\frac{\pi }{2}\end{array}$ is continuous at $x=\frac{\pi }{2}$
Let $\pi -2x=y$
$x=\frac{\pi }{2}-\frac{y}{2}$
as $x\to \frac{\pi }{2},y\to 0$
$={lim}_{y\to 0}\frac{k\cos (\frac{\pi }{2}-\frac{y}{2})}{y}={lim}_{y\to 0}\frac{k\sin \frac{y}{2}}{y}={lim}_{y\to 0}\frac{k\sin \frac{y}{2}}{\frac{y}{2}}\times \frac{1}{2}=\frac{k}{2}$
$f\left(\frac{\pi }{2}\right)={lim}_{x\to \frac{\pi }{2}}f\left(x\right)3=\frac{k}{2}$
$k=6$