Why is $ 2n^2\log n+3n^2 \notin \Omega(n^3)$?

Question

Why $ 2n^2\;log\;n+3n^2 \notin \Omega(n^3)$ ? We can choose constant $c=1$ and $n\ge1$ and the inequality will be true.

Answer 1

When compared to polynomials, logarithms are vastly smaller. For instance, $\log^{100} x < x^{1/100}$ for sufficiently large $x$.

So for loose asymptotics, it's not unreasonable to "ignore logs." So morally, you are asking if $n^2 > c n^3$ for some positive constant $c$ and sufficiently large $n$, and this is clearly not the case.

Another common replacement is to replace $\log x$ by $x^{\epsilon}$ for some really small $\epsilon$. If you want to be concrete, perhaps $\epsilon = 0.01$. Then you're asking about $n^{2.01} > c n^3$, which is also clearly not the case for large $n$.