Задача №2003

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