Đáp án:
\[\lim \frac{{{5^{n - 1}} - {3^n} + {{24}^{\frac{n}{2}}}}}{{{5^n} + {{2.3}^{n + 1}}}} = \frac{1}{5}\]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
\lim \frac{{{5^{n - 1}} - {3^n} + {{24}^{\frac{n}{2}}}}}{{{5^n} + {{2.3}^{n + 1}}}}\\
= \lim \frac{{{5^{n - 1}} - {3^n} + {{\sqrt {24} }^n}}}{{{5^n} + {{2.3}^{n + 1}}}}\\
= \lim \frac{{\frac{1}{5} - {{\left( {\frac{3}{5}} \right)}^n} + {{\left( {\frac{{\sqrt {24} }}{5}} \right)}^n}}}{{1 + 6.{{\left( {\frac{3}{5}} \right)}^n}}}\\
0 < \frac{3}{5} < 1 \Rightarrow \lim {\left( {\frac{3}{5}} \right)^n} = 0\\
0 < \frac{{\sqrt {24} }}{5} < 1 \Rightarrow \lim {\left( {\frac{{\sqrt {24} }}{5}} \right)^n} = 0\\
\Rightarrow \lim \frac{{\frac{1}{5} - {{\left( {\frac{3}{5}} \right)}^n} + {{\left( {\frac{{\sqrt {24} }}{5}} \right)}^n}}}{{1 + 6.{{\left( {\frac{3}{5}} \right)}^n}}} = \frac{{\frac{1}{5} - 0 + 0}}{{1 + 6.0}} = \frac{1}{5}\\
\Rightarrow \lim \frac{{{5^{n - 1}} - {3^n} + {{24}^{\frac{n}{2}}}}}{{{5^n} + {{2.3}^{n + 1}}}} = \frac{1}{5}
\end{array}\)
