Get an alternate form of an equation 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

$n + \frac{1 - n}{n^2 +1}$?
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:
$$
\frac{n^3+1}{n^2+1}=\frac{n^3+n-n+1}{n^2+1}=
\frac{n^3+n}{n^2+1}+\frac{1-n}{n^2+1}=
n+\frac{1-n}{n^2+1}
$$
by using the long division ..................