Вычислить предел:1.) lim(tg^3(x)-3tgx)/cos(x+pi/6) при x стремящемся к pi/3.

[latex]\lim_{x\to\frac{\pi}{3}}\frac{\tan^3 x-3\tan x}{\cos(x+\frac{\pi}{6})}=[/latex] [latex]=\lim_{x\to\frac{\pi}{3}}\frac{\tan x(\tan^2 x-3)}{\cos(x+\frac{\pi}{6})}=\lim_{x\to\frac{\pi}{3}}\frac{\tan x(\tan x+\sqrt{3})*(\tan x-\sqrt{3})}{\cos(x+\frac{\pi}{6})}=[/latex] по свойству пределов [latex]\lim_{x\to c}(a*b)=\lim_{x\to c}a*\lim_{x\to c}b[/latex] [latex]=\lim_{x\to\frac{\pi}{3}}\tan x(\tan x+\sqrt{3})*\lim_{x\to\frac{\pi}{3}}\frac{(\tan x-\sqrt{3})}{\cos(x+\frac{\pi}{6})}=[/latex] [latex]=\tan \frac{\pi}{3}(\tan \frac{\pi}{3}+\sqrt{3})*\lim_{x\to\frac{\pi}{3}}\frac{(\tan x-\sqrt{3})}{\cos(x+\frac{\pi}{6})}=[/latex] [latex]=\sqrt{3}(\sqrt{3}+\sqrt{3})*\lim_{x\to\frac{\pi}{3}}\frac{(\tan x-\sqrt{3})}{\cos(x+\frac{\pi}{6})}=[/latex] [latex]=\sqrt{3}*2\sqrt{3}*\lim_{x\to\frac{\pi}{3}}\frac{(\tan x-\sqrt{3})}{\cos(x+\frac{\pi}{6})}=[/latex] [latex]=6*\lim_{x\to\frac{\pi}{3}}\frac{(\tan x-\sqrt{3})}{\cos(x+\frac{\pi}{6})}=[/latex] Замена [latex]t=x-\frac{\pi}{3}[/latex]  при [latex]x\to\frac{\pi}{3}[/latex]. Тогда [latex]t\to 0.[/latex] [latex]x=t+\frac{\pi}{3}[/latex] [latex]=6*\lim_{t\to 0}\frac{\tan (t+\frac{\pi}{3})-\sqrt{3}}{\cos(t+\frac{\pi}{3}+\frac{\pi}{6})}=[/latex] [latex]=6*\lim_{t\to 0}\frac{\tan (t+\frac{\pi}{3})-\sqrt{3}}{\cos(t+\frac{\pi}{2})}=[/latex] Преобразуем знаменатель. Получим косинус суммы cos (a+b)=cos a*cos b-sin a sin b [latex]\cos(t+\frac{\pi}{2})=\cos t\cos\frac{\pi}{2}-\sin t\sin\frac{\pi}{2}=\cos t*0-\sin t*1=-\sin t[/latex] Перепишем предел в новом виде [latex]=6*\lim_{t\to 0}\frac{\tan (t+\frac{\pi}{3})-\sqrt{3}}{-\sin t}=6*\lim_{t\to 0}\frac{\tan (t+\frac{\pi}{3})-\tan\frac{\pi}{3}}{-\sin t}=[/latex] Воспользуемся формулой разности тангенсов, чтобы преобразовать числитель [latex]\tan(a-b)=\frac{\sin(a-b)}{\cos a\cos b}[/latex] [latex]=6*\lim_{t\to 0}\frac{\tan (t+\frac{\pi}{3})-\tan\frac{\pi}{3}}{-\sin t}=[/latex] [latex]=6*\lim_{t\to 0}\frac{\tan (t+\frac{\pi}{3})-\tan\frac{\pi}{3}}{-\sin t}=6*\lim_{t\to 0}\frac{\sin(t+\frac{\pi}{3}-\frac{\pi}{3})}{\cos(t+\frac{\pi}{3})\cos\frac{\pi}{3}}*\frac{1}{-\sin t}=[/latex] [latex]=6*\lim_{t\to 0}\frac{\sin t}{\cos(t+\frac{\pi}{3})\cos\frac{\pi}{3}}*\frac{1}{-\sin t}=[/latex] [latex]=6*\lim_{t\to 0}\frac{\sin t}{\frac{1}{2}*\frac{1}{2}}*\frac{1}{-\sin t}=6*4\lim_{t\to 0}\frac{\sin t}{-\sin t}=-24.[/latex]