Cho biết \(\lim \dfrac{{{5^n} - 1}}{{{3^n} + 1}}\) bằng

* Hướng dẫn giải

\(\lim \dfrac{{{5^n} - 1}}{{{3^n} + 1}} = \lim \dfrac{{1 - {{\left( {\dfrac{1}{5}} \right)}^n}}}{{{{\left( {\dfrac{3}{5}} \right)}^n} + {{\left( {\dfrac{1}{5}} \right)}^n}}}\)

Do \(\lim \left( {1 - {{\left( {\dfrac{1}{5}} \right)}^n}} \right) = 1 > 0\), \(\lim \left( {{{\left( {\dfrac{3}{5}} \right)}^n} + {{\left( {\dfrac{1}{5}} \right)}^n}} \right) = 0\)và \({\left( {\dfrac{3}{5}} \right)^n} + {\left( {\dfrac{1}{5}} \right)^n} > 0\)nên

\(\lim \dfrac{{1 - {{\left( {\dfrac{1}{5}} \right)}^n}}}{{{{\left( {\dfrac{3}{5}} \right)}^n} + {{\left( {\dfrac{1}{5}} \right)}^n}}} =  + \infty \)