I am not relieved of the following problem and trip.
Let $\varepsilon>0$ be a small parameter, $a>0$ be a given constant, $x_{\varepsilon}\in(0,a)$ be a given sequence such that $x_{\varepsilon}\to a$ as $\varepsilon\to0$ and $f:[0,a)\times[0,a)\to\mathbb{R}$ be a continuous function such that $$ f(x,x)=+\infty\ \text{for all $x\in[0,a)$},\quad f(x,y)>0\ \text{for all $(x,y)\in[0,a)\times[0,x)$} $$ and $$ \lim_{y\to a}\lim_{\varepsilon\to0}f(x_{\varepsilon},y)=+\infty,\quad \lim_{\varepsilon\to0}f(x_{\varepsilon},y)>0\quad\text{for all $y\in[0,a)$} $$ and $y\mapsto \lim_{\varepsilon\to0}f(x_{\varepsilon},y)$ is continuous function on $[0,a)$.
Answer if or not the followings hold: $$ (1) \lim_{\varepsilon\to0}\int_{0}^{x_{\varepsilon}-\varepsilon}f(x_{\varepsilon},y)dy\ge0 $$ $$ (2) \lim_{\varepsilon\to0}\int_{0}^{x_{\varepsilon}-\varepsilon}f(x_{\varepsilon},y)dy=\int_{0}^{a}f(a,y)dy $$
Of course, if (2) is true, (1) automatically holds. However, I think that (2) does not be expected although I don't know whether it holds. The typical example of $f$ is $$ f(x,y)=\frac{g(x,y)}{(x-y)^{3/2}}, $$ where $g$ has good properties, for example, it is continuous on compact sets.
I'm glad if you give solution or counterexample if exists.