Rational sequence/Convergence/R/Example

We consider the sequence given by

${}x_{n}={\frac {-5n^{3}+6n^{2}-n+8}{11n^{3}+7n^{2}+3n-1}}\,,$

and want to know whether it converges and if so, what the limit is. We can not use fact immediately, as neither the numerator nor the denominator converges. However, we can use the following trick. We write

${}x_{n}={\frac {-5n^{3}+6n^{2}-n+8}{11n^{3}+7n^{2}+3n-1}}={\frac {{\left(-5n^{3}+6n^{2}-n+8\right)}{\frac {1}{n^{3}}}}{{\left(11n^{3}+7n^{2}+3n-1\right)}{\frac {1}{n^{3}}}}}={\frac {-5+{\frac {6}{n}}-{\frac {1}{n^{2}}}+{\frac {8}{n^{3}}}}{11+{\frac {7}{n}}+{\frac {3}{n^{2}}}-{\frac {1}{n^{3}}}}}\,.$

In this form, the numerator and the denominator converges, and the limits are ${}-5$ and ${}11$ respectively. Therefore, the sequence converges to ${}-{\frac {5}{11}}$.