I've been studying special functions from Askey,Roy and in proof of Gauss's summation formula (theorem 2.2.2) it states that $\lim_{n\to \infty}\frac{\Gamma (c+n-a)\Gamma(c+n-b)}{\Gamma (c+n) \Gamma(c+n-a-b)}\ _2F_1(a,b;c+n;1)=1$. Only thing we know about $a,b,c$ is that $Re(c-a-b)>0$
I dont understand how is that correct. I have tryed using $lim_{n\to\infty}\frac{\Gamma (n+a)}{\Gamma(n)n^a}=1$ but I dont know what to do with hypergeometric function.