Задача №1993

\[ \lim_{x\to{0}}\frac{\cos{x}-\cos{3x}}{x^2}=\left|\frac{0}{0}\right| =\lim_{x\to{0}}\frac{-2\sin{2x}\cdot\sin(-x)}{x^2} =\lim_{x\to{0}}\frac{2\sin{2x}\cdot\sin{x}}{x^2}=\\ =\lim_{x\to{0}}\frac{4\sin^2x\cdot\cos{x}}{x^2} =4\cdot\lim_{x\to{0}}\left(\left(\frac{\sin{x}}{x}\right)^2\cdot\cos{x}\right)=4\cdot{1^2}\cdot{1}=4. \]