How to prove this $\theta$ notation

Question

How do you prove that the folowing function is equal to $\theta(n^2)$?

$$f(n)=\frac{n^3+n+1}{2n+\ln(n)}.$$

Then $f(n)=\theta(n^2)$.

Thanks!

There should be some value $\theta$ to prove that $\theta(n^2)$ is bigger and smaller than the function f(n) — Jody, Oct 25 '14 at 11:45

score 0 · Accepted Answer · answered Oct 25 '14 at 11:44

0

Hint: Consider $f(n)/n^2$ and take the limit where $n\to \infty$

answered Oct 25 '14 at 11:44

Belgi

Thanks! That did the trick. Solving this will result in $0.5(n^2)$ – Jody Oct 25 '14 at 12:02

score 0 · Answer 2 · answered Oct 25 '14 at 11:54

0

Hints:

$\lim_{n\to \infty}f(n)/n^2=\lim_{n\to \infty}\frac{n+\frac1n+\frac{1}{n^2}}{2n+\ln n}=\frac12$

answered Oct 25 '14 at 11:54

Paul

2 Answers2