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Question
\[\lim_{x \to \pi/2} \frac{2^{- \cos x} - 1}{x\left( x - \frac{\pi}{2} \right)}\]
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Solution
\[ = \lim_{x \to \frac{\pi}{2}} \left[ \frac{2^{- \sin \left( \frac{\pi}{2} - x \right)} - 1}{x\left( x - \frac{\pi}{2} \right)} \right] \left\{ \because \cos x = \sin \left( \frac{\pi}{2} - x \right) \right\}\]
= `\lim_{x \to \frac{\pi}{2}} \left[ \frac{2^\sin \left( x - \frac{\pi}{2} \right) - 1}{\left( x - \frac{\pi}{2} \right) \times x} \right]`
=` \lim_{x \to \frac{\pi}{2}} \left[ \frac{2^\sin \left( x - \frac{\pi}{2} \right) - 1}{\sin \left( x - \frac{\pi}{2} \right)} \times \frac{\sin \left( x - \frac{\pi}{2} \right)}{\left( x - \frac{\pi}{2} \right)} \times \frac{1}{x} \right]`
\[ = \log_e 2 \times 1 \times \frac{1}{\frac{\pi}{2}}\]
\[ = \frac{2}{\pi} \log_e 2\]
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