How little o can verify for one example?

Question

I have following facts:

$n^2log_2n+n^2$ = $o(n^3)$

how I can understand this is true fact? I'm not rain into a homework question, any intuitively idea to quickly find or doing by math?

Answer 1

Just check if $\frac{n^2\log_2(n)}{n^3}+\frac{n^2}{n^3}$ tends to zero as $n$ goes to infinity. This can be easily checked. We know that $$\frac{n^2\log_2(n)}{n^3}+\frac{n^2}{n^3}=\frac{\log_2{n}}{n}+\frac{1}{n}$$ $1/n$ tends to 0, it is a basic property of limits, which can be verfied computationally, and $\frac{\log_2(n)}{n}$ can be verified by L'Hôpital's rule.
In general, $f(x)=o(g(x))$ if $$\lim_{n\to\infty}\frac{f(n)}{g(n)}=0$$ This expression means that as we put larger and larger values of $n$ in $f(n)/g(n)$, the result gets closer and closer to 0. This is the mathematical definition, but it is equivalent to the computer science definition.