Limit Equals Infinity for a Sequence

Question

I have to solve the following question in my assignment:

Prove that: $\lim\limits_{n \to \infty} \frac{n^2-n}{n+2}$ = $\infty$.

I have to prove this with the following definition:

"A series $(a_n)^\infty_{n=1}$ approches infinity if for every real number ${M}$, a natural number ${N}$ such that for every ${n}>{N}$, ${a_n}>{M}$".

I know how to solve the question with limit arithmetic, however, with the definition I have to work with, it's harder.

Your help is appreciated, thank you.

*Sorry for the bad mathjax, I am a newbie with the system.

Answer 1

$$\lim\limits_{n\to\infty}\frac{n^2-n}{n+2}\color{red}{>}\lim\limits_{n\to\infty}\frac{n^2-n\color{red}{-6}}{n+2}=\lim\limits_{n\to\infty}\frac{(n+2)(n-3)}{n+2}=\lim\limits_{n\to\infty}n-3=\infty$$

Answer 2

Hint:

$$ \frac{n^2-n}{n+2}\ge \frac{n^2-n}{2n}=\frac{1}{2}(n-1) $$ with $n\ge 2$

Answer 3

Can you show that, for every natural number $n\ge 2$, that $\frac{n-1}{n+2} \ge \frac 14$? This means that $\frac{n^2 - n}{n+2} = \frac{n(n-1)}{n+2} \ge \frac n 4$ if $n \ge 2$, so that $$n > 4N \implies \frac{n(n-1)}{n+2} > N$$ for any natural number $N$.

Answer 4

$$\frac{n^2-n}{n+2}\ge\frac{n^2-n}{2n}=\frac n2-\frac12>M\iff n>2M+1$$