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Limits and Derivatives — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsLimits and Derivatives5 Marks
Question
Evaluate the following limits.
$\lim\limits_{\text{x} \rightarrow\pi}\frac{1-\sin\frac{\text{x}}{2}}{\cos\frac{\text{x}}{2}\Big(\cos\frac{\text{x}}{4}-\sin\frac{\text{x}}{4}\Big)}$ 
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Answer

Given $\lim\limits_{\text{x} \rightarrow\pi}\frac{1-\sin\frac{\text{x}}{2}}{\cos\frac{\text{x}}{2}\Big(\cos\frac{\text{x}}{4}-\sin\frac{\text{x}}{4}\Big)}$
$=\lim\limits_{\text{x} \rightarrow\pi}\frac{\cos^{2}\frac{\text{x}}{4}+\sin^{2}\frac{\text{x}}{4}-2\sin\frac{\text{x}}{4}\cdot\cos\frac{\text{x}}{4}}{\cos\frac{\text{x}}{2}\Big(\cos\frac{\text{x}}{4}-\sin\frac{\text{x}}{4}\Big)\Big(\cos\frac{\text{x}}{4}-\sin\frac{\text{x}}{4}\Big)}$
$=\lim\limits_{\text{x} \rightarrow\pi}\frac{\Big(\cos\frac{\text{x}}{4}-\sin\frac{\text{x}}{4}\Big)^{2}}{\cos\frac{\text{x}}{2}\Big(\cos\frac{\text{x}}{4}-\sin\frac{\text{x}}{4}\Big)\Big(\cos\frac{\text{x}}{4}-\sin\frac{\text{x}}{4}\Big)\Big(\cos\frac{\pi}{4}-\sin\frac{\text{x}}{4}\big)}$
$=\lim\limits_{\text{x} \rightarrow\pi}\frac{1}{\Big(\cos\frac{\pi}{4}+\sin\frac{\pi}{4}\Big)}$
$=\frac{1}{\cos\frac{\pi}{4}+\sin\frac{\pi}{4}}$
$=\frac{1}{\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}}=\frac{1}{\frac{2}{\sqrt{2}}}$
$=\frac{1}{\sqrt{2}}$
Hence. the required answer is $\frac{1}{\sqrt{2}}.$

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