[tex]\LARGE\begin{aligned}&\lim_{n\to\infty}\left(\dfrac{2n+2}{2n+1}\right)^{10n}=\\&\lim_{n\to\infty}\left(\dfrac{2n+1+1}{2n+1}\right)^{10n}=\\&\lim_{n\to\infty}\left(1+\dfrac{1}{2n+1}\right)^{10n}=\\&\lim_{n\to\infty}\left(\left(1+\dfrac{1}{2n+1}\right)^{2n+1}\right)^{\frac{10n}{2n+1}}=\\&\lim_{n\to\infty}e^{\frac{10n}{2n+1}}=\\&e^{\displaystyle\lim_{n\to\infty}\frac{10n}{2n+1}}=\\&e^{\displaystyle\lim_{n\to\infty}\frac{10}{2+\frac{1}{n}}}=\\&e^{\frac{10}{2}}=e^5\end[/tex]
