Question

How to simplify this (don't need to find the limit).

Answer 1

\[\lim_{n \rightarrow \infty}\frac{ \sqrt{n+1}(1+n^2) }{ \sqrt{n}[1+(n+1)^2] }\]

Answer 2

\[\lim_{n \rightarrow \infty}\frac{ \sqrt{n} (n)^2}{ \sqrt{n}(n+1)^2 }\] This is the simplified version, but not sure how to get it like that.

Answer 3

I wouldn't really call it simplifying. It's more using the fact that \(\sqrt{n+1}\approx\sqrt n\) for very large \(n\). Similarly, \(1+n^2\approx n^2\) and \(1+(n+1)^2\approx(n+1)^2\) for large \(n\).