Question

Evaluate the following limits:

$\underset{x\to \mathrm{\pi }}{\lim }\frac{1-\sin {\displaystyle \frac{x}{2}}}{\cos {\displaystyle \frac{x}{2}}\left(\cos {\displaystyle \frac{x}{4}}-\sin {\displaystyle \frac{x}{4}}\right)}$ [NCERT EXEMPLAR]

Answer 1

$\underset{x\to \mathrm{\pi }}{\lim }\frac{1-\sin {\displaystyle \frac{x}{2}}}{\cos {\displaystyle \frac{x}{2}}\left(\cos {\displaystyle \frac{x}{4}}-\sin {\displaystyle \frac{x}{4}}\right)}$

Put $x=\mathrm{\pi }+h$

When $x\to \mathrm{\pi },h\to 0$

$$\begin{gathered} \therefore \underset{x\to \mathrm{\pi }}{\lim }\frac{1-\sin {\displaystyle \frac{x}{2}}}{\cos {\displaystyle \frac{x}{2}}\left(\cos {\displaystyle \frac{x}{4}}-\sin {\displaystyle \frac{x}{4}}\right)} \\ =\underset{h\to 0}{\lim }\frac{1-\sin \left({\displaystyle \frac{\mathrm{\pi }+h}{2}}\right)}{\cos \left({\displaystyle \frac{\mathrm{\pi }+h}{2}}\right)\left[\cos \left({\displaystyle \frac{\mathrm{\pi }+h}{4}}\right)-\sin \left({\displaystyle \frac{\mathrm{\pi }+h}{4}}\right)\right]} \\ =\underset{h\to 0}{\lim }\frac{1-\sin \left({\displaystyle \frac{\mathrm{\pi }}{2}+\frac{h}{2}}\right)}{\cos \left({\displaystyle \frac{\mathrm{\pi }}{2}+\frac{h}{2}}\right)\left[\cos \left({\displaystyle \frac{\mathrm{\pi }}{4}+\frac{h}{4}}\right)-\sin \left({\displaystyle \frac{\mathrm{\pi }}{4}+\frac{h}{4}}\right)\right]} \\ =\underset{h\to 0}{\lim }\frac{1-\cos \left({\displaystyle \frac{h}{2}}\right)}{-\sin \left({\displaystyle \frac{h}{2}}\right)\left[\left(\cos {\displaystyle \frac{\mathrm{\pi }}{4}}\cos {\displaystyle \frac{h}{4}}-\sin {\displaystyle \frac{\mathrm{\pi }}{4}}\sin {\displaystyle \frac{h}{4}}\right)-\left(\sin {\displaystyle \frac{\mathrm{\pi }}{4}}\cos {\displaystyle \frac{h}{4}}+\cos {\displaystyle \frac{\mathrm{\pi }}{4}}\sin {\displaystyle \frac{h}{4}}\right)\right]} \end{gathered}$$
$$\begin{gathered} =\underset{h\to 0}{\lim }\frac{2{\sin }^2{\displaystyle \frac{h}{4}}}{-2\sin {\displaystyle \frac{h}{4}}\cos {\displaystyle \frac{h}{4}}\left({\displaystyle \frac{1}{\sqrt{2}}}\cos {\displaystyle \frac{h}{4}}-{\displaystyle \frac{1}{\sqrt{2}}}\sin {\displaystyle \frac{h}{4}}-{\displaystyle \frac{1}{\sqrt{2}}}\cos {\displaystyle \frac{h}{4}}-{\displaystyle \frac{1}{\sqrt{2}}}\sin {\displaystyle \frac{h}{4}}\right)} \\ =\underset{h\to 0}{\lim }\frac{\sin {\displaystyle \frac{h}{4}}}{-\cos {\displaystyle \frac{h}{4}}\times \left(-\sqrt{2}\sin {\displaystyle \frac{h}{4}}\right)} \\ =\frac{1}{\sqrt{2}}\times \frac{1}{\underset{h\to 0}{\lim }\cos {\displaystyle \frac{h}{4}}} \\ =\frac{1}{\sqrt{2}}\times 1\left(\underset{\theta \to 0}{\lim }\cos \theta =1\right) \\ =\frac{1}{\sqrt{2}} \end{gathered}$$