Tính các giới hạn sau:

\(\lim \frac{{{4^n} + {5^n}}}{{{{3.4}^n} - {{4.5}^n}}}\).

\(\lim \frac{{{4^n} + {5^n}}}{{{{3.4}^n} - {{4.5}^n}}}\)\( = \lim \frac{{{5^n}\left( {\frac{{{4^n}}}{{{5^n}}} + 1} \right)}}{{{5^n}\left( {\frac{{{{3.4}^n}}}{{{5^n}}} - 4} \right)}}\)\( = \lim \frac{{{{\left( {\frac{4}{5}} \right)}^n} + 1}}{{3.{{\left( {\frac{4}{5}} \right)}^n} - 4}}\)

\( = \frac{{\lim \left[ {{{\left( {\frac{4}{5}} \right)}^n} + 1} \right]}}{{\lim \left[ {3.{{\left( {\frac{4}{5}} \right)}^n} - 4} \right]}} = \frac{1}{{ - 4}} = - \frac{1}{4}\).