[tex]\huge\boxed{\lim_{n\to\infty}\left(1+\dfrac{1}{n}\right)^n=e}[/tex]
1.
[tex]\huge\begin{aligned}&\lim_{n\to\infty}\left(1+\dfrac{2}{n}\right)^n=\\&=\lim_{n\to\infty}\left(\left(1+\dfrac{2}{n}\right)^{\frac{n}{2}}\right)^2=\\&=\left(\lim_{n\to\infty}\left(1+\dfrac{2}{n}\right)^{\frac{n}{2}}\right)^2=e^2\end{alignned}[/tex]
2.
[tex]\huge\begin{aligned}&\lim_{n\to\infty}\left(1-\dfrac{2}{n^2}\right)^n=\\&=\lim_{n\to\infty}\left(\left(1+\left(-\dfrac{2}{n^2}\right)\right)^{-\frac{n^2}{2}}\right)^{-\frac{2}{n}}=\\&=\left(\lim_{n\to\infty}\left(1+\left(-\dfrac{2}{n^2}\right)\right)^{-\frac{n^2}{2}}\right)^{-\frac{2}{n}}=\\&=e\,^{\lim\limits_{_{n\to\infty}}-\frac{2}{n}}}=e^0=1\end{aligned}}[/tex]
