Question

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$lim_{x to 1} (x-1)tg}frac{pi*x}{2}$

Answer 1

$displaystyle lim_{x to 1}(x-1){rm tg}frac{pi x}{2}=left{begin{array}{ccc}x-1=t\x=t+1\tto 0end{array}right}=lim_{t to 0}bigg[tcdot{rm tg}bigg(frac{pi t}{2}+frac{pi}{2}bigg)bigg]=\ \ \ =-lim_{t to 0}bigg(tcdot {rm ctg}frac{pi t}{2}bigg)=-lim_{t to 0}frac{tcosfrac{pi t}{2}}{sin frac{pi t}{2}}=-lim_{t to 0}frac{t}{sin frac{pi t}{2}}=-lim_{t to 0}frac{tcdot frac{pi}{2}}{frac{pi}{2}cdot sin frac{pi t}{2}}=-frac{2}{pi}$