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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.
$\lim\limits_{\text{x} \rightarrow\frac{\pi}{4}}\frac{\tan^{3}\text{x}-\tan\text{x}}{\cos\Big(\text{x}+\frac{\pi}{4}\Big)}$ 
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Answer

Given$\lim\limits_{\text{x} \rightarrow\frac{\pi}{4}}\frac{\tan^{3}\text{x}-\tan\text{x}}{\cos\Big(\text{x}+\frac{\pi}{4}\Big)}$
$=\lim\limits_{\text{x} \rightarrow\frac{\pi}{4}}\frac{\tan\text{x}(\tan^{2}\text{x}-1)}{\cos\Big(\text{x}+\frac{\pi}{4}\Big)}$
$=\lim\limits_{\text{x} \rightarrow\frac{\pi}{4}}\tan\text{x}\cdot\lim\limits_{\text{x} \rightarrow\frac{\pi}{4}}\Bigg[\frac{(1-\tan^{2}\text{x})}{\cos\Big(\text{x}+\frac{\pi}{4}\Big)}\Bigg]$
$=-1\times\lim\limits_{\text{x} \rightarrow\frac{\pi}{4}}\frac{(1-\tan\text{x})(1+\tan\text{x})}{\cos\Big(\text{x}+\frac{\pi}{4}\Big)}$
$=-(1+1)\times\lim\limits_{\text{x} \rightarrow\frac{\pi}{4}}\frac{(\cos\text{x}-\sin\text{x})}{\cos\text{x}\cdot\cos\Big(\text{x}+\frac{\pi}{4}\Big)}$
$=-2\times\lim\limits_{\text{x} \rightarrow\frac{\pi}{4}}\frac{\sqrt{2}\Big(\frac{1}{\sqrt{2}}\cos\text{x}-\frac{1}{\sqrt{2}}\sin\text{x}\Big)}{\cos\text{x}\cdot\cos\Big(\text{x}+\frac{\pi}{4}\Big)}$
$=-2\sqrt{2}\lim\limits_{\text{x} \rightarrow\frac{\pi}{4}}\frac{\Big[\cos\frac{\pi}{4}\cdot\cos\text{x}-\sin\frac{\pi}{4}\sin\text{x}\Big]}{\cos\text{x}\cdot\cos\Big(\text{x}+\frac{\pi}{4}\Big)}$
$=\lim\limits_{\text{x} \rightarrow\frac{\pi}{4}}\frac{-2\sqrt{2}\cdot\cos\Big(\text{x}+\frac{\pi}{4}\Big)}{\cos\text{x}\cdot\cos\Big(\text{x}+\frac{\pi}{4}\Big)}$
$=\frac{-2\sqrt{2}}{\cos\frac{\pi}{4}}$
$=\frac{-2\sqrt{2}}{\frac{1}{\sqrt{2}}}=-2\times2=-4$
Hence, the rquired answer is -4.

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