Evaluate the following limits: _ x 4 [ ^2 x- ^2 x x-cosec x ]

Question

Evaluate the following limits: $\lim _{x \rightarrow \frac{\pi}{4}}\left[\frac{\tan ^2 x-\cot ^2 x}{\sec x-\operatorname{cosec} x}\right]$

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Answer

$ \lim _{x \rightarrow \frac{\pi}{4}} \frac{\tan ^2 x-\cot ^2 x}{\sec x-\operatorname{cosec} x}$
$=\lim _{x \rightarrow \frac{\pi}{4}} \frac{\left(\sec ^2 x-1\right)-\left(\operatorname{cosec}^2 x-1\right)}{\sec x-\operatorname{cosec} x}$
$=\lim _{x \rightarrow \frac{\pi}{4}} \frac{\sec ^2 x-\operatorname{cosec}^2 x}{\sec x-\operatorname{cosec} x}$
$=\lim _{x \rightarrow \frac{\pi}{4}} \frac{(\sec x-\operatorname{cosec} x)(\sec x+\operatorname{cosec} x)}{(\sec x-\operatorname{cosec} x)}$
$=\lim _{x \rightarrow \frac{\pi}{4}}(\sec x+\operatorname{cosec} x)\cdots\left[\begin{array}{l}
\because x \rightarrow \frac{\pi}{4}, \sec x \rightarrow \sqrt{2} \text { and } \operatorname{cosec} x \rightarrow \sqrt{2} \\
\therefore \sec x-\operatorname{cosec} x \rightarrow 0, \therefore \sec x-\operatorname{cosec} x \neq 0
\end{array}\right]$
$=\lim _{x \rightarrow \frac{\pi}{4}}(\sec x)+\lim _{x \rightarrow \frac{\pi}{4}}(\operatorname{cosec} x)$
$=\sec \frac{\pi}{4}+\operatorname{cosec} \frac{\pi}{4}$
$=\sqrt{2}+\sqrt{2}=2 \sqrt{2} $

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