Solve this
Question:

If $f(x)=\left\{\begin{array}{cc}\frac{\sin (\cos x)-\cos x}{(\pi-2 x)^{2}}, & x \neq \frac{\pi}{2} \\ k & , x=\frac{\pi}{2}\end{array}\right.$ is continuous at $x=\pi / 2$, then $k$ is equal to

(a) 0

(b) $\frac{1}{2}$

(C) 1

(d) $-1$

Solution:

(a) 0

Given: $f(x)=\left\{\begin{array}{l}\frac{\sin (\cos x)-\cos x}{(\pi-2 \mathrm{x})^{2}}, x \neq \frac{\pi}{2} \\ k, x=\frac{\pi}{2}\end{array}\right.$

If $f(x)$ is continuous at $x=\frac{\pi}{2}$, then

$\lim _{x \rightarrow \frac{\pi}{2}} f(x)=f\left(\frac{\pi}{2}\right)$

$\Rightarrow \lim _{x \rightarrow \frac{\pi}{2}} \frac{\sin (\cos x)-\cos x}{(\pi-2 \mathrm{x})^{2}}=k$

Now,

$\frac{\pi}{2}-x=y$

$\Rightarrow \pi-2 x=2 y$

Also, $x \rightarrow \frac{\pi}{2}, y \rightarrow 0$

$\Rightarrow \lim _{y \rightarrow 0} \frac{\sin \left(\cos \left(\frac{\pi}{2}-y\right)\right)-\cos \left(\frac{\pi}{2}-y\right)}{4 y^{2}}=k$

$\Rightarrow \lim _{y \rightarrow 0} \frac{\sin (\sin y)-\sin (y)}{4 \mathrm{y}^{2}}=k$               $\left[\because \sin C-\sin D=2 s\right.$ in $\left.\left(\frac{C-D}{2}\right) \cos \left(\frac{C+D}{2}\right)\right]$

$\Rightarrow \frac{1}{2} \lim _{y \rightarrow 0} \frac{\sin \left(\frac{\sin y-y}{2}\right)}{y} \frac{\cos \left(\frac{\sin y+y}{2}\right)}{y}=k$

$\Rightarrow \frac{1}{2} \lim _{y \rightarrow 0} \frac{\left(\frac{\sin y-y}{2}\right) \sin \left(\frac{\sin y-y}{2}\right)}{y\left(\frac{\sin y-y}{2}\right)} \frac{\cos \left(\frac{\sin y+y}{2}\right)}{y}=k$

$\Rightarrow \frac{1}{2} \lim _{y \rightarrow 0}\left(\frac{\left(\frac{\sin y-y}{2}\right)}{y}\right)\left(\frac{\sin \left(\frac{\sin y-y}{2}\right)}{\left(\frac{\sin y-y}{2}\right)}\right)\left(\frac{\cos \left(\frac{\sin y+y}{2}\right)}{y}\right)=k$

$\Rightarrow \frac{1}{2} \lim _{y \rightarrow 0}\left(\frac{\left(\frac{\sin y-y}{2}\right)}{y}\right) \lim _{y \rightarrow 0}\left(\frac{\sin \left(\frac{\sin y-y}{2}\right)}{\left(\frac{\sin y-y}{2}\right)}\right) \lim _{y \rightarrow 0}\left(\frac{\cos \left(\frac{\sin y+y}{2}\right)}{y}\right)=k$

$\Rightarrow \frac{1}{4} \lim _{y \rightarrow 0}\left(\frac{\sin y}{y}-1\right) \lim _{y \rightarrow 0}\left(\frac{\sin \left(\frac{\sin y-y}{2}\right)}{\left(\frac{\sin y-y}{2}\right)}\right) \lim _{y \rightarrow 0}\left(\frac{\cos \left(\frac{\sin y+y}{2}\right)}{y}\right)=k$

$\Rightarrow \frac{1}{4} \times 0 \times 1 \times \lim _{y \rightarrow 0}\left(\frac{\cos \left(\frac{\sin y+y}{2}\right)}{y}\right)=k$

$\Rightarrow 0=k$

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