Limit of incomplete beta function

Question

Consider the incomplete beta function $I_p(rn, n) = \frac{1}{B(rn, n)}\int_{0}^{p}t^{rn-1}(1-t)^{n-1}dt$, where $r, p \in (0, 1)$ are constants and $B(rn, n) = \int_{0}^1 t^{rn-1}(1-t)^{n-1}dt$ is the beta function.

1. Does the limit of $I_p(rn, n)$ as $n$ goes to $\infty$ exist? if it exists, what’s the limit?