Proving Big Omega of a polynomial without limits
Define $M = \max(|a_0|/a_m, |a_1|/a_m, \ldots, |a_{m-1}|/a_m)$, and take $c = a_m/2$ and $n_0 = 2mM$. Then for $n \geq n_0$,
$$
\begin{align*}
f(n) &amp;= a_m n^m \left(1 + \frac{a_{m-1}}{a_m} \cdot \frac{1}{n} + \cdots + \frac{a_1}{a_m} \cdot \frac{1}{n^{m-1}} + \frac{a_0}{a_m} \cdot \frac{1}{n^m}\right) \\ &amp;\geq
a_m n^m \left(1 - \frac{M}{n_0} - \cdots - \frac{M}{n_0^{m-1}} - \frac{M}{n_0^M}\right) \\ &amp;\geq
a_m n^m \left(1 - \frac{mM}{n_0}\right) \\ &amp;=
c n^m.
\end{align*}
$$