Задача №1182

\[ \lim_{x\to{0}}\frac{1-\cos(1-\cos{x})}{x^4} =\left[\frac{0}{0}\right] =\lim_{x\to{0}}\frac{\left(1-\cos(1-\cos{x})\right)\cdot\left(1+\cos(1-\cos{x})\right)}{x^4\cdot\left(1+\cos(1-\cos{x})\right)}=\\ =\lim_{x\to{0}}\frac{\sin^2(1-\cos{x})}{x^4\cdot\left(1+\cos(1-\cos{x})\right)} =\lim_{x\to{0}}\left(\left(\frac{\sin\left(2\sin^2\frac{x}{2}\right)}{2\sin^2\frac{x}{2}}\right)^2\cdot\left(\frac{\sin\frac{x}{2}}{\frac{x}{2}}\right)^4\cdot\frac{1}{4\cdot\left(1+\cos(1-\cos{x})\right)}\right) =\frac{1}{8}. \]