Let $(Z_1,Z_2)$ be a bivariate standard normal vector and $x\in\mathbb{R}$. We consider $$f(\sigma_l):=\left| \operatorname{E}[1\{Z_1\leq x/\sigma_l\}1\{Z_2\leq x/\sigma_l\}]-\operatorname{E}[1\{Z_1\leq x\}1\{Z_2\leq x\}]\right|,$$ where $\sigma_l=1+ O(l^s)$, $s\in \mathbb{R}_{<0}$.
Can we say something about how fast $f(\sigma_l)$ tends to $0$ if $l\rightarrow\infty$? Something like $f(\sigma_l)\in O(l^s)$? Or is it possible to show, that
$$g(u):=\operatorname{E}[1\{Z_1\leq x/u\}1\{Z_2\leq x/u\}]$$ with $u\in R>0$ is differentiable on $[\sigma_l,1]$ (w.l.o.g. $\sigma_l<1$) as a function of $u$ so that we could apply the Mean value therem?
Let $h(x)=P[Z_1\leqslant x,Z_2\leqslant x]$. For every nonnegative $y$, $$h(x+y)\leqslant h(x)+P[Z_1\in (x,x+y)]+P[Z_2\in (x,x+y)]. $$ The density of a normal random variable $Z$ with variance $\sigma^2$ is uniformly bounded by $$ c(\sigma^2)=\frac1{\sqrt{2\pi\sigma^2}}, $$ hence, for every Borel set $B$, $P[Z_k\in B]\leqslant c(\mathrm{var}(Z_k))\cdot|B|$. Finally, for every $x$ and $y$, $$ |h(x)-h(y)|\leqslant(c(\mathrm{var}(Z_1))+c(\mathrm{var}(Z_2)))\cdot|x-y|. $$