[tex]\displaystyle \limit\lim_{n\to\infty} e_n=\displaystyle \limit\lim_{n\to\infty} (\sqrt{n^2+2n}-n)^n=\displaystyle \limit\lim_{n\to\infty} \left(\dfrac{n^2+2n-n^2}{\sqrt{n^2+2n}+n}\right)^n=\\ =\displaystyle \limit\lim_{n\to\infty} \left(\dfrac{ 2n}{\sqrt{n^2+2n}+n}\right)^n \stackrel{1^{\infty}}{=} \displaystyle \limit\lim_{n\to\infty} \left(1+\dfrac{n-\sqrt{n^2+2n}}{\sqrt{n^2+2n}+n}\right)^n= \\ \stackrel{\text{calcule}}{===} \displaystyle \limit\lim_{n\to\infty} e ^{n\cdot \frac{n-\sqrt{n^2+2n}}{\sqrt{n^2+2n}+n}} \stackrel{\text{calcule}}{===} \boxed{e^{\frac{-1}{2}}} [/tex]
