Đáp án:
\[\lim \frac{{{{\left( { - 4} \right)}^n} - {2^n}}}{{{{5.3}^{n + 1}} - {{7.5}^n}}} = 0\]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
\lim \frac{{{{\left( { - 4} \right)}^n} - {2^n}}}{{{{5.3}^{n + 1}} - {{7.5}^n}}}\\
= \lim \frac{{{{\left( { - \frac{4}{5}} \right)}^n} - {{\left( {\frac{2}{5}} \right)}^n}}}{{15.{{\left( {\frac{3}{5}} \right)}^n} - 7}}\\
\left. \begin{array}{l}
\lim {\left( { - \frac{4}{5}} \right)^n} = 0\\
\lim {\left( {\frac{2}{5}} \right)^n} = 0
\end{array} \right\} \Rightarrow \lim \left[ {{{\left( { - \frac{4}{5}} \right)}^n} - {{\left( {\frac{2}{5}} \right)}^n}} \right] = 0\\
\lim \left[ {15.{{\left( {\frac{3}{5}} \right)}^n}} \right] = 0 \Rightarrow \lim \left[ {15.{{\left( {\frac{3}{5}} \right)}^n} - 7} \right] = - 7\\
\Rightarrow \lim \frac{{{{\left( { - \frac{4}{5}} \right)}^n} - {{\left( {\frac{2}{5}} \right)}^n}}}{{15.{{\left( {\frac{3}{5}} \right)}^n} - 7}} = 0\\
\Rightarrow \lim \frac{{{{\left( { - 4} \right)}^n} - {2^n}}}{{{{5.3}^{n + 1}} - {{7.5}^n}}} = 0
\end{array}\)
