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Find the values of K. So that the function $ f(x) = \left\{ \begin{array}{l l} \large\frac{k \cos x}{\pi -2x} & \quad if \quad x \neq \large\frac{\pi}{2} \\ 3 & \quad if \quad x =\large\frac{\pi}{2} \end{array} \right. $ is continuous at $x =\large\frac{\pi}{2}$ .

$\begin{array}{1 1}(A)\;6 \\(B)\;4 \\(C)\;5 \\(D)\;8 \end{array}$

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Since $f(x)$ is continuous at $x= \large\frac{\pi}{2}$
$f(\large\frac{\pi}{2})$$=\lim \limits_{x \to \large\frac{\pi}{2} } f(x) =\lim \limits _{x \to \large\frac{\pi}{2}} \large=\frac{K \cos x }{\pi-2x}$
Here we need not find left hand and right hand separately because $f(x) $ is not different when $ x < \large\frac{\pi}{2} $ and $ x > \large\frac{\pi}{2} $
=> $ \lim \limits_{h \to 0} \large\frac{k \cos (\large\frac{\pi}{2} +h)}{\pi -2 (\large\frac{\pi}{2} +h)}$
Putting $x =\large\frac{\pi}{2} $$+h$
So that $x \to \large\frac{\pi}{2} $$ h \to 0$
$\qquad= \lim \limits _{h \to 0} \large\frac{-K \sin h}{-2h}$
$\qquad= \large\frac{K}{2}$$ \lim \limits_{h \to 0} \bigg( \large\frac{\sin h}{h}\bigg)$
$\qquad= \large\frac{k}{2}$$.1$
$\qquad= \large\frac{k}{2}$
Hence A is the correct answer.
answered Apr 22, 2014 by meena.p

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