[m]\lim_{x \to\infty}(\frac{5x+1}{5x-4})^{12x-5}=\lim_{x \to\infty}(\frac{5x+1}{5x-4})^{12x}\cdot (\frac{5x+1}{5x-4})^{-5}=[/m]
Предел произведения равен произведению пределов:
[m]\lim_{x \to\infty}(\frac{5x+1}{5x-4})^{12x}\cdot\lim_{x \to\infty}(\frac{5x+1}{5x-4})^{-5}=[/m]
[m]=\lim_{x \to\infty}(\frac{5x+1}{5x-4})^{12x}\cdot 1^{-5}=
\lim_{x \to\infty}(\frac{5x+1}{5x-4})^{12x}=[/m]
Делим и числитель и знаменатель дроби на 5х:
[m]=\lim_{x \to\infty}(\frac{\frac{5x+1}{5x}}{\frac{5x-4}{5x}})^{12x} =[/m]
[m]\lim_{x \to\infty}\frac{(1+\frac{1}{5x})^{12x}}{(1-\frac{1}{5x})^{12x}}=[/m]
[m]\lim_{x \to\infty}\frac{((1+\frac{1}{5x})^{5x})^{\frac{12}{5}}}{((1-\frac{4}{5x})^{-5x})^{\frac{-12}{5}}}=\frac{e^{\frac{12}{5}}}{e^{\frac{-12}{5}}}=e^{\frac{12}{5}-(-\frac{12}{5})}=e^{4,8}[/m]