Using graphing software of the regularized incomplete beta function I notice that for $I_{x}(a,b)$ where $a = k + 1$ and $b = n - k$. Where $n = Ck$ for $C \in \mathbb{N}$ and $C \geq 2$ we get a step function where $I_{x}(a,b)$ is equal to $0$ for $x < \frac{k}{n}$ and $1$ otherwise. I would like to prove
\begin{equation} \lim_{k \rightarrow \infty} I_{x}(k+1,n-k) =\lim_{k \rightarrow \infty} \frac{ \int^{x}_{0} t^{k}(1-t)^{n-k-1}dt}{\int^{1}_{0} t^{k}(1-t)^{n-k-1}dt}= 0 \end{equation} for $x < \frac{k}{n}$
I have tried some asymptotic expansions and approximations of the beta functions and the incomplete beta function such as $B_{x}(a,b) \sim \frac{x^a}{a}(1+\mathcal{O}(x))$ however this yields some very incorrect results (such as values that tend towards infinity when we know the regularized incomplete beta function is bounded between $0$ and $1$ for values of $x$ between $0$ and $1$.