[tex]\text{Se folosesc urmatoarele formule:}[/tex]
[tex]\lim\limits_{n\to a}\left(1+\dfrac{1}{u(n)}\right)^{u(n)} = e,\quad \text{cand }u(n) \to \infty[/tex]
[tex]e^{i\pi}+1 = 0 \Rightarrow e^{i\pi} = -1\Big|\sqrt{}\Rightarrow \sqrt{e^{i\pi}}=\sqrt{-1}\Rightarrow e^{\dfrac{i\pi}{2}} = i[/tex]
[tex]4.\,\,\,\lim\limits_{n\to \infty}\left(1+i\dfrac{\pi}{2n}\right)^n = \lim\limits_{n\to \infty}\left(\left(1+\dfrac{1}{\dfrac{2}{i\pi}\cdot n}\right)^{\dfrac{2}{i\pi}\cdot n}\right)^{\dfrac{i\pi}{2}} = \\ =e^{\dfrac{i\pi}{2}}=\boxed{i}[/tex]
[tex]5.\,\,\,\lim\limits_{n\to \infty}\left(1+\dfrac{1+\pi i}{n}\right)^n = \lim\limits_{n\to \infty}\left(\left(1+\dfrac{1}{\dfrac{n}{1+\pi i}}\right)^{\dfrac{n}{1+\pi i}}\right)^{1+\pi i}=[/tex]
[tex]=e^{1+\pi i} = e\cdot e^{\pi i} = e\cdot (-1) = \boxed{-e}[/tex]
