I am looking at this questions and the proof for it and wondering how this works.Can anyone explain the answer to me or do you have any other way to answer this question.I am new to asymptotic notations
$$2n^3 + 5n + 12 = O(n^3)$$
True Proof: $$\lim_{n\to\infty}\frac{2n^3 + 5n + 12}{n^3}=2$$
This implies $$2n^3 + 5n + 12 = O(n^3)$$
A function $f(n)$ is $O(g(n))$ if there is some constant $k$ such that $f(n)\leq k\cdot g(n)$. I prefer directly prove that $2n^3+5n+12=O(n^3)$ as $2n^3+5n+12\leq 2n^3+5n^3+12n^3=19n^3$.