Question

Помогите решить, используя второй замечательный предел

Answer 1

24

$\lim\limits_{x \to 0} (1 + tg^2(\sqrt{x}))^\frac{3}{x} = \lim\limits_{x \to 0} (1 + x)^\frac{3}{x} = \lim\limits_{x \to 0} (((1 + x)^{\frac{1}{x}})^x)^\frac{3}{x} = \lim\limits_{x \to 0} e^\frac{3x}{x} = e^3$

25

$\lim\limits_{x \to 0} (\cos(x))^\frac{1}{\sin(x)} = \lim\limits_{x \to 0} (1 - (1 - \cos(x)))^\frac{1}{\sin(x)} = \lim\limits_{x \to 0} (1 - 2\sin^2(x/2))^{\frac{1}{\sin(x)}} =\\= \lim\limits_{x \to 0} (1 - \frac{x^2}{2})^{\frac{1}{x}} = \lim\limits_{x \to 0} (((1 - \frac{x^2}{2})^\frac{2}{x^2})^\frac{x^2}{2})^{\frac{1}{x}} = \lim\limits_{x \to 0} e^{\frac{x^2}{2x}} = 1$