Its straightforward to see that $lim_{n\rightarrow \infty}\frac{n^{m-1}}{n^m} = 0$, $m$ is a fixed positive integer. This meas that $n^m$ grows faster that $n^{m-1}$.
Now, let be $\{a_0,a_1,...,a_m\}$ rational numbers, consider the following limit:
$lim_{n\rightarrow \infty} a_0+a_1 n+a_2 n^2+...+a_m n^m$.
It would seems to me that this goes to $+\infty$ or $-\infty$. Since $a_m n^m$ grows (decreases) faster than any other summand, we just check the sign of $a_m $.
How can I prove this? Any suggestion?
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