Question

Вычислите lim x->pi/2 (sinx)^tgx через формулу lim f (x)^g(x)=e^lim g(x) ln f(x)

Answer 1

Ответ:

1

Пошаговое объяснение:

$\displaystyle \lim_{x\to\frac{\pi}{2}}\sin x^{\tan x}=\exp(\lim_{x\to\frac\pi2}\tan x\ln\sin x)=\exp(\lim_{x\to\frac\pi2}\dfrac{\tan x}{\ln\sin x^{-1}})=\exp(\lim_{x\to\frac\pi2}\dfrac{\tan x'}{\ln\sin x^{-1}'})=\exp(\lim_{x\to\frac\pi2}\dfrac{\frac{1}{\cos^2x}}{-\frac{\cos x}{\ln\sin x^2*\sin x}})=\exp(-\lim_{x\to\frac\pi2}\frac{\ln\sin x^2*\sin x}{\cos^3x})=\exp(-\lim_{x\to\frac\pi2}\frac{\ln\sin\frac\pi2^2*\sin\frac\pi2}{\cos\frac\pi2^3}=\exp(-\frac01)=\exp0=1$