If function f is defined such that f(x)= cc k x -2 x , & if x 2 3… - Vidyadip

Question

If function $f$ is defined such that
$
f(x)=\left\{\begin{array}{cc}
\frac{k \cos x}{\pi-2 x}, & \text { if } x \neq \frac{\pi}{2} \\
3, & \text { if } x=\frac{\pi}{2}
\end{array}\right.
$is continuous at $x=\frac{\pi}{2}$, then value of $k$ :

✓

Answer

(C)
$
\begin{aligned}
\lim _{x \rightarrow \frac{\pi}{2}} f(x) & =\lim _{x \rightarrow \frac{\pi}{2}} \frac{k \cos x}{\pi-2 x} \\
& =\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} \\
& =\frac{k}{2} \lim _{k \rightarrow 0} \frac{\sin h}{h} \\
& =\frac{k}{2}(1)=\frac{k}{2}
\end{aligned}
$
for continuity at $x=\frac{\pi}{2}$
$
\begin{aligned}
& & \lim _{x \rightarrow \frac{\pi}{2}} f(x) & =f\left(\frac{\pi}{2}\right)\\
\Rightarrow & & \frac{k}{2} & =3 \\
\therefore & & k & =6
\end{aligned}
$

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