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Question
The value of \[\lim_{x \to \pi/2} \left( \sec x - \tan x \right)\]is
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Solution
0
\[ = \lim_{h \to 0} \left( \sec \left( \frac{\pi}{2} - h \right) - \tan \left( \frac{\pi}{2} - h \right) \right)\]
\[ = \lim_{h \to 0} \left( cosec h - \cot h \right)\]
\[ = \lim_{h \to 0} \frac{1 - \cos h}{\sin h}\]
\[ = \lim_{h \to 0} \frac{2 \sin^2 \frac{h}{2}}{\sin h}\]
\[ = \lim_{h \to 0} \frac{2 \sin^2 \frac{h}{2}}{2 \sin \frac{h}{2}\cos \frac{h}{2}}\]
\[ = \lim_{h \to 0} \tan \frac{h}{2}\]
\[ = 0\]
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