Real numbers/Rational sequences/Convergence behavior/Exercise

Let ${}P=\sum _{i=0}^{d}a_{i}x^{i}$ and ${}Q=\sum _{i=0}^{e}b_{i}x^{i}$ be polynomials with ${}a_{d},b_{e}\neq 0$. Determine, depending on ${}d$ and ${}e$, whether

${}z_{n}={\frac {P(n)}{Q(n)}}\,$

(which is defined for ${}n$ sufficiently large) is a convergent sequence or not, and determine the limit in the convergent case.

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