Задача №1182
Условие
Найти предел \(\lim_{x\to{0}}\frac{1-\cos(1-\cos{x})}{x^4}\).
Решение
\[
\lim_{x\to{0}}\frac{1-\cos(1-\cos{x})}{x^4}
=\left[\frac{0}{0}\right]
=\lim_{x\to{0}}\frac{\left(1-\cos(1-\cos{x})\right)\cdot\left(1+\cos(1-\cos{x})\right)}{x^4\cdot\left(1+\cos(1-\cos{x})\right)}=\\
=\lim_{x\to{0}}\frac{\sin^2(1-\cos{x})}{x^4\cdot\left(1+\cos(1-\cos{x})\right)}
=\lim_{x\to{0}}\left(\left(\frac{\sin\left(2\sin^2\frac{x}{2}\right)}{2\sin^2\frac{x}{2}}\right)^2\cdot\left(\frac{\sin\frac{x}{2}}{\frac{x}{2}}\right)^4\cdot\frac{1}{4\cdot\left(1+\cos(1-\cos{x})\right)}\right)
=\frac{1}{8}.
\]
Ответ:
\(\frac{1}{8}\)