Question

Вычислить пределы функций, используя второй замечательный предел:

Answer 1

$1) limlimits _{x to infty}Big (frac{4x^2-3}{4x^2+1}Big )^{1-2x^2}=limlimits _{x to infty}Big (Big (1+frac{-4}{4x^2+1}Big )^{frac{4x^2+1}{-4}}Big )^{frac{-4(1-2x^2)}{4x^2+1}}=\\=e^{limlimits _{x to infty}frac{8x^2-4}{4x^2+1} }=e^{frac{8}{4} }=e^2$

$2); ; limlimits _{x to infty}Big (frac{3x^3+7}{3x^3-1}Big )^{1-x^3}=limlimits _{x to infty}Big (Big (1+frac{8}{3x^3-1}Big )^{frac{3x^3-1}{8}}Big )^{frac{8(1-x^3)}{3x^2-1}}=\\=e^{limlimits _{x to infty}frac{-8x^2+8}{3x^2-1} }=e^{-frac{8}{3} }$

$3); ; limlimits _{x to infty}Big (frac{6x-7}{6x+20}Big )^{x-3}=limlimits _{x to infty}Big (Big (1+frac{-27}{6x+20}Big )^{frac{6x+20}{-27} }Big )^{frac{-27(x-3)}{6x+20} }=\\=e^{ limlimits _{x to infty}frac{-27x+81}{6x+20} }=e^{-frac{27}{6} }=e^{-frac{9}{2} }$

$4); ; limlimits _{x to infty}Big (frac{3x^2-4}{3x^2-10}Big )^{2x^2+1}=limlimits _{x to infty}Big (Big (1+frac{6}{3x^2-10}Big )^{frac{3x^2-10}{6} }Big )^{frac{6(2x^2+1)}{3x^2-10} }=e^{4}$