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Question
\[\lim_\theta \to \pi/2 \frac{1 - \sin \theta}{\left( \pi/2 - \theta \right) \cos \theta}\] is equal to
Options
1
−1
\[\frac{1}{2}\]
\[- \frac{1}{2}\]
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Solution
\[\frac{1}{2}\]
\[ = \lim_{h \to 0} \frac{1 - \cos h}{\left( \frac{\pi}{2} - \left( \frac{\pi}{2} - h \right) \right) \sin h}\]
\[ = \lim_{h \to 0} \frac{2 \sin^2 \frac{h}{2}}{h \sin h}\]
\[ = \lim_{h \to 0} \frac{2 \sin^2 \frac{h}{2}}{\frac{\frac{4 h^2}{4}}{\frac{\sin h}{h}}}\]
\[ = \frac{2}{4}\]
\[ = \frac{1}{2}\]
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