Question

$\displaystyle\lim _{x \rightarrow \pi} \frac{1-\sin \frac{x}{2}}{\left(\cos \frac{x}{2}\right)\left(\frac{\cos x}{4}-\frac{\sin x}{4}\right)}=$

• A. $\frac{3}{\sqrt{2}}$
• B. $\frac{-1}{\sqrt{2}}$
• C. $\frac{1}{\sqrt{2}}$
• D. $\frac{5}{\sqrt{2}}$

Answer 1

Answer: (C)

$\displaystyle\lim _{x \rightarrow \pi} \frac{1-\sin \frac{x}{2}}{\left(\cos \frac{x}{2}\right)\left(\cos \frac{x}{4}-\sin \frac{x}{4}\right)}$
Let $x=\pi+h, x \rightarrow \pi, h \rightarrow 0$
$=\displaystyle\lim _{x \rightarrow \pi} \frac{1-\sin \left(\frac{\pi}{2}+\frac{h}{2}\right)}{\cos \left(\frac{\pi}{2}+\frac{h}{2}\right)\left(\cos \left(\frac{\pi+h}{4}\right)-\sin \left(\frac{\pi+h}{4}\right)\right)}$
$=\displaystyle\lim _{h \rightarrow 0} \frac{1-\cos \frac{h}{2}}{-\sin \frac{h}{2} \cdot\left(\cos \frac{\pi}{4} \cos \frac{h}{4}-\sin \frac{\pi}{4} \sin \frac{h}{4}-\sin \frac{\pi}{4} \cdot \cos \frac{h}{4}-\cos \frac{\pi}{4} \cdot \sin \frac{h}{4}\right.}$
$=\displaystyle\lim _{h \rightarrow 0} \frac{\sqrt{2}\left(1-\cos \frac{h}{2}\right)}{-\sin \frac{h}{2}\left\{\cos \frac{h}{4}-\sin \frac{h}{4}-\cos \frac{h}{4}-\sin \frac{h}{4}\right\}}$
$=\displaystyle\lim _{h \rightarrow 0} \frac{\sqrt{2}\left(1-\cos \frac{h}{2}\right)}{\sin \frac{h}{2} \cdot\left(-2 \sin \frac{h}{4}\right)}$
$=\displaystyle\lim _{h \rightarrow 0} \frac{\sqrt{2} \cdot 2 \sin \frac{2 h}{4}}{2 \cdot \sin \frac{h}{2} \times \sin \frac{h}{4}}$
$=\displaystyle\lim _{h \rightarrow 0} \sqrt{2} \frac{\sin \frac{h}{4}}{\sin \frac{h}{2}}=\sqrt{2} \lim _{h \rightarrow 0} \frac{\sin \frac{h}{4}}{4\left(\frac{h}{4}\right)} \cdot \frac{2(h / 2)}{\sin \frac{h}{2}}$
$=\sqrt{2} \times \frac{2}{4} \cdot \displaystyle\lim _{h \rightarrow 0}\left(\frac{\sin \frac{h}{4}}{\frac{h}{4}}\right)\left(\frac{\frac{h}{2}}{\sin \frac{h}{2}}\right)$
$=\sqrt{2} \times \frac{2}{4} \times 1 \times 1=\frac{1}{\sqrt{2}}$