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Let $f(x)=\left\{\begin{array}{cc}\frac{k \cos x}{\pi-2 x} & \text { when } x \neq \frac{\pi}{2} \text { and if } \\ 3 & x=\frac{\pi}{2}\end{array}\right.$ in $\lim _{x \rightarrow \frac{\pi}{2}} f(x)=f\left(\frac{\pi}{2}\right)$, find the value of $k$
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Verified Answer
$$
\begin{aligned}
&f\left(\frac{\pi}{2}\right)=\lim _{x \rightarrow \frac{\pi}{2}} f(x)=\lim _{x \rightarrow \frac{\pi}{2}} \frac{k \cos x}{\pi-2 x} \\
&\Rightarrow \quad 3=\lim _{h \rightarrow 0} \frac{k \cos \left(\frac{\pi}{2}+h\right)}{\pi-2\left(\frac{\pi}{2}+h\right)}=\lim _{h \rightarrow 0} \frac{k(-\sin h)}{-2 h} \\
&\Rightarrow \quad 3=\lim _{h \rightarrow 0} \frac{k}{2}\left(\frac{\sin h}{h}\right) \Rightarrow 3=\frac{k}{2} \Rightarrow k=6
\end{aligned}
$$
\begin{aligned}
&f\left(\frac{\pi}{2}\right)=\lim _{x \rightarrow \frac{\pi}{2}} f(x)=\lim _{x \rightarrow \frac{\pi}{2}} \frac{k \cos x}{\pi-2 x} \\
&\Rightarrow \quad 3=\lim _{h \rightarrow 0} \frac{k \cos \left(\frac{\pi}{2}+h\right)}{\pi-2\left(\frac{\pi}{2}+h\right)}=\lim _{h \rightarrow 0} \frac{k(-\sin h)}{-2 h} \\
&\Rightarrow \quad 3=\lim _{h \rightarrow 0} \frac{k}{2}\left(\frac{\sin h}{h}\right) \Rightarrow 3=\frac{k}{2} \Rightarrow k=6
\end{aligned}
$$
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