Question

Помогите решить, используя второй замечательный предел

Answer 1

24

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

25

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