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Question
\[\lim_{x \to \frac{\pi}{2}} \left( \frac{\pi}{2} - x \right) \tan x\]
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Solution
\[\lim_{x \to 1} \frac{1 - \frac{1}{x}}{\sin \pi \left( x - 1 \right)}\]
\[ = \lim_{x \to 1} \frac{x - 1}{x \sin \pi\left( x - 1 \right)}\]
\[Let y = x - 1\]
\[ x \to 1\]
\[ \therefore y \to 0\]
\[ = \lim_{y \to 0} \frac{y}{\left( y + 1 \right) \sin \pi y}\]
\[ = \lim_{y \to 0} \frac{1}{\pi\left( y + 1 \right) \times \frac{\sin \pi y}{\pi y}}\]
\[ = \frac{1}{\pi\left( 0 + 1 \right) \times 1}\]
\[ = \frac{1}{\pi}\]
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