Задача №1130

\[ \lim_{x\to\pi}\frac{1-\sin\frac{x}{2}}{\cos\frac{x}{2}\cdot\left(\cos\frac{x}{4}-\sin\frac{x}{2}\right)} =\left|\frac{0}{0}\right| =\lim_{x\to\pi}\frac{\left(1-\sin\frac{x}{2}\right)\cdot\left(1+\sin\frac{x}{2}\right)\cdot\left(\cos\frac{x}{4}+\sin\frac{x}{4}\right)}{\cos\frac{x}{2}\cdot\left(\cos\frac{x}{4}-\sin\frac{x}{4}\right)\cdot\left(\cos\frac{x}{4}+\sin\frac{x}{4}\right)\cdot\left(1+\sin\frac{x}{2}\right)}=\\ =\lim_{x\to\pi}\frac{\left(1-\sin^2\frac{x}{2}\right)\cdot\left(\cos\frac{x}{4}+\sin\frac{x}{4}\right)}{\cos\frac{x}{2}\cdot\left(\cos^2\frac{x}{4}-\sin^2\frac{x}{4}\right)\cdot\left(1+\sin\frac{x}{2}\right)} =\lim_{x\to\pi}\frac{\cos^2\frac{x}{2}\cdot\left(\cos\frac{x}{4}+\sin\frac{x}{4}\right)}{\cos\frac{x}{2}\cdot\cos\frac{x}{2}\cdot\left(1+\sin\frac{x}{2}\right)}=\\ =\lim_{x\to\pi}\frac{\cos\frac{x}{4}+\sin\frac{x}{4}}{1+\sin\frac{x}{2}} =\frac{\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}}{1+1}=\frac{\sqrt{2}}{2}. \]