IBEasySequences10–12

The Limit of a Rational Sequence

O Limite de uma Sucessão Racional

A sequence is given by $u_{n} = \frac{3n^{2} + 2n}{n^{2} + 5}$ . Find $\lim_{n \to \infty} u_{n}$ and justify why the sequence converges.

Both parts of the fraction grow without bound, so comparing them directly says nothing. Divide through by the highest power present and watch which terms fade.

Solution

Step 1 of 5

1.Divide numerator and denominator by $n^{2}$ :