I saw a some proofs like this:
To prove $N^2+3N=O(N^3)$
$$\lim_{N\to\infty} \frac{N^2+3N}{N^3}=0 $$
then $N^2+3N$ is $O(N^3)$ and if the result is infinity left hand side is Big Omega of right hand side, and There are some situations limits not exits...
I was told this is by some theorem..I can just use it but I would like to know some details and reasoning behind this.
I wonder if some one can give me a link or tell me which aspect to search
thanks for any help!