Question

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

Answer 1

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

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

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