Lim x->oo ((7^x+5)/(7^x+9))^(7^x-5)

Решите задачу:

$\lim\limits _{x \to +\infty} \Big (\frac{7^{x}+5}{7^{x}+9}\Big )^{7^{x}-5} = \lim\limits _{x \to +\infty} \Big (\frac{7^{x}+5}{7^{x}+9}\Big )^{7^{x}}\cdot \Big ( \frac{7^{x}+9}{7^{x}+5} \Big ) ^5=$

$= \lim\limits _{x \to +\infty} \frac{(7^{x}(1+\frac{5}{7^{x}})\, )^{7^{x}}}{(7^{x}\, (1+\frac{9}{7^{x}})\, )^{7^{x}}} \cdot \Big (1+\underbrace {\frac{4}{7^{x}+5}}_{\to 0}}\Big )^5= \lim\limits _{x \to +\infty} \frac{(1+\frac{5}{7^{x}})^{\frac{7^{x}}{5}\cdot 5}}{(1+\frac{9}{7^{x}})^{\frac{7^{x}}{9}}\cdot 9}\cdot 1=$

$= \lim\limits _{x \to +\infty} \frac{e^5}{e^9}= \frac{e^5}{e^9}=\frac{1}{e^4}$