# Formelsammlung Mathematik: Grenzwerte

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##### 1
${\displaystyle \lim _{|\xi |\to \infty }\left(1+{\frac {z}{\xi }}\right)^{\xi }=e^{z}\qquad \forall z\in \mathbb {C} }$

##### 2
${\displaystyle \lim _{n\to \infty }{\frac {n!\,n^{z}}{z\,(z+1)\cdots (z+n)}}=\Gamma (z)\qquad \forall z\notin \mathbb {Z} _{\leq 0}}$

##### 3
${\displaystyle \lim _{n\to \infty }{\frac {1}{n^{y}}}\,\prod _{k=0}^{n}\left(1+{\frac {y}{k+x}}\right)={\frac {\Gamma (x)}{\Gamma (x+y)}}}$

##### 4
${\displaystyle \lim _{\varepsilon \to 0^{+}}\left(\left|\Gamma \left(-n+\varepsilon \,e^{i\varrho _{1}}\right)\right|-\left|\Gamma \left(-n+\varepsilon \,e^{i\varrho _{2}}\right)\right|\right)={\frac {\psi (n+1)}{\Gamma (n+1)}}{\Big (}\cos \varrho _{1}-\cos \varrho _{2}{\Big )}}$
${\displaystyle f\,}$ ${\displaystyle \lim _{z_{1},z_{2}\to 0}{\frac {z_{1}\,f(z_{1})-z_{2}\,f(z_{2})}{z_{1}-z_{2}}}}$
${\displaystyle \Gamma (-n+z)\,}$ ${\displaystyle (-1)^{n}{\frac {\Psi (n+1)}{\Gamma (n+1)}}}$
${\displaystyle \Gamma (-n+z)\,z\,}$ ${\displaystyle (-1)^{n}{\frac {1}{\Gamma (n+1)}}}$
${\displaystyle \Psi (-n+z)\,}$ ${\displaystyle \Psi (n+1)\,}$
${\displaystyle \Psi (-n+z)\,z}$ ${\displaystyle -1\,}$
${\displaystyle \zeta (1+z)\,(1+z)^{\alpha }}$ ${\displaystyle \alpha +\gamma \,}$

##### 5
${\displaystyle \lim _{n\to \infty }{\frac {n!\,e^{n}}{n^{n}\,{\sqrt {2\pi n}}}}=1}$

##### 6
Mit ${\displaystyle z=\varrho e^{i\varphi }}$ gilt ${\displaystyle \Gamma (z)\sim {\sqrt {\frac {2\pi }{z}}}\left({\frac {z}{e}}\right)^{z}\left\{{\begin{matrix}1&-\pi <\varphi <\pi \\{\frac {1}{e^{2\pi iz}-1}}&\varphi =\pi \end{matrix}}\right.}$ für ${\displaystyle \varrho \to \infty }$

##### 7
${\displaystyle \lim _{n\to \infty }{\frac {n}{\sqrt[{n}]{n!}}}=e}$

##### 8
${\displaystyle \lim _{n\to \infty }{\frac {\sqrt {n\pi }}{2^{2n}}}{2n \choose n}=1}$

##### 9
${\displaystyle B(\alpha n,\beta n)\sim {\sqrt {\frac {2\pi }{n}}}\,{\sqrt {\frac {\alpha +\beta }{\alpha \,\beta }}}\left({\frac {\alpha ^{\alpha }\,\beta ^{\beta }}{(\alpha +\beta )^{\alpha +\beta }}}\right)^{n}}$

##### 10
${\displaystyle \lim _{n\to \infty }{\sqrt[{n}]{\frac {(\alpha n+\beta n)!}{(\alpha n)!\,(\beta n)!}}}={\frac {(\alpha +\beta )^{\alpha +\beta }}{\alpha ^{\alpha }\,\beta ^{\beta }}}}$

##### 11
${\displaystyle \lim _{y\to \infty }{\frac {|\Gamma (x+iy)|}{{\sqrt {\frac {2\pi }{y}}}\,y^{x}\,e^{-{\frac {\pi }{2}}y}}}=1\qquad x\in \mathbb {R} }$

##### 12
${\displaystyle A:=\lim _{n\to \infty }{\frac {e^{\frac {n^{2}}{4}}\,\prod _{k=1}^{n}k^{k}}{n^{{\frac {n^{2}}{2}}+{\frac {n}{2}}+{\frac {1}{12}}}}}=e^{{\frac {1}{12}}-\zeta '(-1)}}$

##### 13
${\displaystyle \lim _{n\to \infty }\left[{\frac {(n+1)^{n+1}}{n^{n}}}-{\frac {n^{n}}{(n-1)^{n-1}}}\right]=e}$

##### 14
${\displaystyle \lim _{m\to \infty }\left(\sum _{k=1}^{m}{\frac {(\log k)^{n}}{k}}-{\frac {(\log m)^{n+1}}{n+1}}\right)=\gamma _{n}}$

##### 15
${\displaystyle \lim _{n\to \infty }\left(H_{n}-\log n\right)=\gamma }$

##### 16
${\displaystyle \lim _{n\to \infty }\left(\log n-\sum _{k=0}^{n}{\frac {1}{k+x}}\right)=\psi (x)}$

##### 17
${\displaystyle \lim _{z\to 1}\left(\zeta (z)-{\frac {1}{z-1}}\right)=\gamma }$

##### 18
${\displaystyle \lim _{n\to \infty }{\frac {1}{n!}}\int _{n+a{\sqrt {n}}}^{n+b{\sqrt {n}}}{\frac {x^{n}}{e^{x}}}\,dx={\frac {1}{\sqrt {2\pi }}}\,\int _{a}^{b}e^{-{\frac {x^{2}}{2}}}\,dx\qquad a,b\in \mathbb {C} }$