Given $y>x>0$, and $x$, $y \sim N(0,1)$. Show
$$\lim_{x \to \infty, y \to x} \frac{\frac{x+y}{2}-\frac{2}{\frac{1}{x}+\frac{1}{y}}}{\frac{x+y}{2}-\frac{\phi(x)-\phi(y)}{\Phi(y)-\Phi(x)}}=0$$
where $\phi$ is the density, and $\Phi$ is the distribution of the standard normal.
I have tried that as $y \to x$, $$\frac{\phi(x)-\phi(y)}{\Phi(y)-\Phi(x)} \to \frac{\phi'(x)}{-\Phi'(x)}=\frac{\phi(x)(-x)}{-\phi(x)}=x$$, and don't know how to proceed. Any idea would be helpful!