Comparing growth of two sums of functions

Question

Does $n+n^4$ grow faster than $n^2+n^3$? If so, why?

Answer 1

Yes, asymptotically. You can see that $n + n^4 = \omega(n^3 + n^2)$ by taking the limit:

$$ \lim_{n \to \infty} \frac{n^4 +n}{n^3 + n^2} = \lim_{n \to \infty}\frac{n + n^{-2}}{1 + n^{-1}} = + \infty $$.

See also this answer.

Answer 2

1. Polynomials of higher degree grow faster than polynomials of lower degree.
2. The polynomial $n + n^4$ is of degree 4.
3. The polynomial $n^2 + n^3$ is of degree 3.
4. Therefore, the $n^4$ polynomial grows faster than the other polynomial.

Answer 3

Let me add way with simple inequalities: $$n^2+n^3 \leqslant n^4 +n^4=2n^4 \leqslant 2(n^4+n) $$ Now taking constant $C=2$ we have $n^2+n^3 \in O(n^4+n)= O(n^4)$.