(1)

\begin{align*} \Gamma(x) & =\int_{0}^{\infty}t^{x-1}e^{-t}dt\\ & =\lim_{n\rightarrow\infty}\int_{0}^{n}t^{x-1}(1-\frac{t}{n})^{n}dt\\ & =\lim_{n\rightarrow\infty}n^{x}\int_{0}^{1}s^{x-1}(1-s)^{n}ds\qquad,\qquad t=ns\\ & =\lim_{n\rightarrow\infty}n^{x}B(n+1,x)\\ & =\lim_{n\rightarrow\infty}n^{x}\frac{\Gamma(n+1)\Gamma(x)}{\Gamma(x+n+1)}\\ & =\lim_{n\rightarrow\infty}n^{x}n!P^{-1}(x+n,n+1)\\ & =\lim_{n\rightarrow\infty}n^{x}n!Q^{-1}(x,n+1) \end{align*}

(2)

\begin{align*} \Gamma(x) & =\lim_{n\rightarrow\infty}n^{x}n!P(x,-(n+1))\\ & =\lim_{n\rightarrow\infty}n^{x}n!\prod_{j=0}^{n}\frac{1}{x+j}\\ & =\frac{1}{x}\lim_{n\rightarrow\infty}\prod_{k=1}^{n-1}\left(\frac{k+1}{k}\right)^{x}\prod_{j=1}^{n}\frac{j}{x+j}\\ & =\frac{1}{x}\prod_{k=1}^{\infty}\left(1+\frac{1}{k}\right)^{x}\prod_{j=1}^{\infty}\left(1+\frac{x}{j}\right)^{-1} \end{align*}

(3)

\begin{align*} \frac{1}{\Gamma(x)} & =\lim_{n\rightarrow\infty}\frac{n^{-x}}{n!}P^{-1}(x,-(n+1))\\ & =\lim_{n\rightarrow\infty}\frac{n^{-x}}{n!}\prod_{j=0}^{n}(x+j)\\ & =x\lim_{n\rightarrow\infty}n^{-x}\prod_{j=1}^{n}\left(\frac{x+j}{j}\right)\\ & =x\lim_{n\rightarrow\infty}\exp(-x\log n)\prod_{j=1}^{n}\left(1+\frac{x}{j}\right)\exp\left(\frac{x}{j}\right)\exp\left(-\frac{x}{j}\right)\\ & =x\lim_{n\rightarrow\infty}\exp(-x\log n)\exp\left(x\sum_{j=1}^{n}\frac{1}{j}\right)\prod_{j=1}^{n}\left(1+\frac{x}{j}\right)\exp\left(-\frac{x}{j}\right)\\ & =x\lim_{n\rightarrow\infty}\exp\left\{ x\left(\sum_{j=1}^{n}\frac{1}{j}-\log n\right)\right\} \prod_{j=1}^{n}\left(1+\frac{x}{j}\right)\exp\left(-\frac{x}{j}\right)\\ & =xe^{x\gamma}\prod_{j=1}^{\infty}\left(1+\frac{x}{j}\right)e{}^{-\frac{x}{j}} \end{align*}