Задача №1137

\[ \lim_{h\to{0}}\frac{\tg(a+2h)-2\tg(a+h)+\tg{a}}{h^2} =\left[\frac{0}{0}\right] =\lim_{h\to{0}}\frac{\tg(a+2h)-\tg(a+h)+\tg{a}-\tg(a+h)}{h^2}=\\ =\lim_{h\to{0}}\left(\frac{\sin{h}}{h^2\cos(a+h)}\cdot\left(\frac{1}{\cos(a+2h)}-\frac{1}{\cos{a}}\right)\right) =\lim_{h\to{0}}\left(\frac{\sin{h}}{h^2\cos(a+h)}\cdot\frac{\cos{a}-\cos(a+2h)}{\cos{a}\cdot\cos(a+2h)}\right)=\\ =\lim_{h\to{0}}\left(\left(\frac{\sin{h}}{h}\right)^2\cdot\frac{2\sin(a+h)}{\cos{a}\cdot\cos(a+h)\cdot\cos(a+2h)}\right) =\frac{2\sin{a}}{\cos^3{a}}. \]