Get an alternate form of an equation

Question

I am trying to prove the following limit value:

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

But how does

$$\frac{n^3 + 1}{n^2 +1}$$

become this

$n + \frac{1 - n}{n^2 +1}$?

Answer 1

by using the long division ..................

Answer 2

Just do the usual math: $$ n+\frac{1-n}{n^2+1}=\frac{n(n^2+1)+1-n}{n^2+1}= \frac{n^3+n+1-n}{n^2+1}=\frac{n^3+1}{n^2+1} $$

If you want to go the other way around: $$ \frac{n^3+1}{n^2+1}=\frac{n^3+n-n+1}{n^2+1}= \frac{n^3+n}{n^2+1}+\frac{1-n}{n^2+1}= n+\frac{1-n}{n^2+1} $$