## Problem Statement

Calculate the limit of the numerical sequence:

$\lim _{n \rightarrow \infty}(n-\sqrt{n(n-1)})$
## Solution

$$
\begin{aligned}
& \lim _{n \rightarrow \infty}(n-\sqrt{n(n-1)})=\lim _{n \rightarrow \infty} \frac{(n-\sqrt{n(n-1)})(n+\sqrt{n(n-1)})}{n+\sqrt{n(n-1)}}= \\
& =\lim _{n \rightarrow \infty} \frac{n^{2}-n(n-1)}{n+\sqrt{n(n-1)}}=\lim _{n \rightarrow \infty} \frac{n^{2}-n^{2}+n}{n+\sqrt{n(n-1)}}= \\
& =\lim _{n \rightarrow \infty} \frac{n}{n+\sqrt{n(n-1)}}=\lim _{n \rightarrow \infty} \frac{\frac{1}{n} n}{\frac{1}{n}(n+\sqrt{n(n-1)}}= \\
& =\lim _{n \rightarrow \infty} \frac{1}{1+\sqrt{1-\frac{1}{n}}}=\frac{1}{1+\sqrt{1-0}}=\frac{1}{2}
\end{aligned}
$$

## Problem Kuznetsov Limits 5-24