Mathematics - Limits Question with Solution | TestHub

${\lim }_{x\to \frac{\pi }{2}}\frac{\tan \left(\frac{\pi }{4}-\frac{x}{2}\right)\cdot (1-\sin x)}{(\pi -2x{)}^3}$ Let $\mathrm{x}=\frac{\pi }{2}+\mathrm{y};\mathrm{y}\to 0$ $={\lim }_{y\to 0}\frac{-\tan \left(-\frac{y}{2}\right)\cdot (1-\cos y)}{(-2y{)}^3}={\lim }_{y\to 0}\frac{-\tan \frac{y}{2}2{\sin }^2\frac{y}{2}}{(-8)\cdot \frac{y^3}{8}\cdot 8}$ $={\lim }_{y\to 0}\frac{1}{32}\frac{\tan \frac{y}{2}}{\left(\frac{y}{2}\right)}\cdot {\left[\frac{\sin y/2}{y/2}\right]}^2=\frac{1}{32}$