Does it converge or diverge? If it converges, then what does it converge to?

Series with n = 1 to infinity $\displaystyle \dfrac{n^{3} - 3}{n^{3} + n^{2} + 1}$

This one would use the limit test, but I can't find$\displaystyle b_{n}$ (The series to compare it to). Any strategies for finding it?

Limit Test is: Limit as n goes toward infinity of $\displaystyle \dfrac{a_{n}}{b_{n}}$

Rules:

If $\displaystyle b_{n} $ is infinity then $\displaystyle a_{n} $ diverges

If$\displaystyle b_{n} = 0 $ then $\displaystyle a_{n} $ converges

If $\displaystyle b_{n} $ comes out to a value, then if $\displaystyle b_{n} $converges, then $\displaystyle a_{n} $ converges, and if $\displaystyle b_{n} $ diverges, then $\displaystyle a_{n} $ diverges.