## Problem Statement

Calculate the limit of the numerical sequence:

$\lim _{n \rightarrow \infty}\left(\frac{2+4+\ldots+2 n}{n+3}-n\right)$
$$
\begin{aligned}
& \lim _{n \rightarrow \infty}\left(\frac{2+4+\ldots+2 n}{n+3}-n\right)=\lim _{n \rightarrow \infty}\left(\frac{\left(\frac{(2+2 n) n}{2}\right)}{n+3}-n\right)= \\
& =\lim _{n \rightarrow \infty}\left(\frac{(1+n) n}{n+3}-n\right)=\lim _{n \rightarrow \infty}\left(\frac{(1+n) n}{n+3}-\frac{n(n+3)}{n+3}\right)= \\
& =\lim _{n \rightarrow \infty} \frac{n+n^{2}-n^{2}-3 n}{n+3}=\lim _{n \rightarrow \infty} \frac{-2 n}{n+3}=\lim _{n \rightarrow \infty} \frac{\frac{1}{n} \cdot(-2) n}{\frac{1}{n}(n+3)}= \\
& =\lim _{n \rightarrow \infty} \frac{-2}{1+\frac{3}{n}}=-\frac{-2}{1+0}=-2
\end{aligned}
$$

## Problem Kuznetsov Limits 6-31