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Question
\[\lim_{x \to 1} \left( 1 - x \right) \tan \left( \frac{\pi x}{2} \right)\]
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Solution
\[\lim_{x \to 1} \left( 1 - x \right) \tan\frac{\pi x}{2}\]
\[ = \lim_{h \to 0} \left\{ 1 - \left( 1 - h \right) \right\} \tan \frac{\pi}{2} \left( 1 - h \right)\]
\[ = \lim_{h \to 0} h \tan \left( \frac{\pi}{2} - \frac{\pi h}{2} \right)\]
\[ = \lim_{h \to 0} h \cot \frac{\pi h}{2}\]
\[ = \lim_{h \to 0} \frac{h}{\tan \frac{\pi h}{2}}\]
\[ = \lim_{h \to 0} \frac{1}{\frac{\tan\frac{\pi h}{2} \times \frac{\pi}{2}}{\frac{\pi h}{2}}}\]
\[ = \frac{1}{\frac{\pi}{2}} \left[ \because \lim_{h \to 0} \tan \frac{h}{h} = 1 \right]\]
\[ = \frac{2}{\pi}\]
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