I am trying to find the limit of an integral of a function for a parameter in that function going to zero.
To be more precise, I am trying to find (EDITED)
$$\lim_{\rho \to 0} \int_0^b ~\Phi \left(\frac {\sqrt{1-\rho} \Phi^{-1}(x) - \Phi^{-1}(c)}{\sqrt \rho} \right) dx,$$
where $c$ is a constant and $\Phi$ is a standard normal cdf with its inverse $\Phi^{-1}$ , so my integrand is continuous and lies between 0 and 1.
EDITED: Equivalently, the above integral can be expressed as the cdf of a bivariate standard normal distribution defined as follows (not sure this is helpful)
$ \frac 1{2\pi \sqrt{\rho}} \int_{-\infty}^{\Phi^{-1}(b)} \int_{-\infty}^{-\Phi^{-1}(c)} exp\left[ - \frac {z^2+2 {\sqrt{1-\rho}} z y + y^2}{2\rho } \right] ~dz~dy $
Basically, I just need to find some justification to exchange the limit and the integral.
I know that normally, one can move the limit into any function, i.e.
$$\lim_{\rho \to 0} g(f(\rho)) = g(\lim_{\rho \to 0}f(\rho)),$$
when $g(\cdot)$ is not dependent on $\rho$, but I am not sure an integral counts as such a function. I have looked at the dominated convergence theorem and the monotone convergence theorem but do not believe I can apply them here because I am looking for the limit going to zero.
As an integral can also be understood as a limit of a sum, the key question might be the interchangeability of the limits, as found in a related post here. However, I am not sure how exactly that applies to my problem!