I am interested in the behavior of the regularized incomplete beta function $I_x(a,\frac{1}{2})$. Graphing on WolframAlpha, I noticed that it behaves similarly to $e^{-ka}$ in the sense that $x\to1$ behaves like $k\to0$.
Here's a graph showing one instance of the similarity. Where does this relationship come from? Why does $I_x(a,\frac{1}{2})$ act like an exponential decay function? And what is the relationship between $x$ and $k$?
I tried checking out some of the representations of $I$ here, but I couldn't figure it out. Any answers would be highly appreciated - thanks a lot!
Edit: Here is a Mathematica Manipulate command which explores this relationship:
Manipulate[Plot[{BetaRegularized[x, (n-1)/2, 1/2], Exp[-(1-x)n]}, {n, 2, 10}], {x, 0, 1}]