fractional Sobolev integral

Question

I am trying to compute this limit $$ \lim_{\varepsilon\rightarrow0^{+}}\frac{1}{\varepsilon^{2}}\int_{0}% ^{e^{-1/\varepsilon}}\int_{0}^{e^{-1/\varepsilon}}\frac{\left\vert \frac {1}{\log x}-\frac{1}{\log y}\right\vert ^{2}}{|x-y|^{2}}dxdy. $$ I would like to understand if the limit is zero. I tried many changes of variables (like $t=-\log x$ and $s=-\log y$) but with no luck. This is taken from a paper, but the computations are skipped, and it is hard to read. An alternative would be to prove $$ \lim_{\varepsilon\rightarrow0^{+}}\frac{1}{\varepsilon^{2}}\int_{0}% ^{e^{-1/\varepsilon}}\int_{0}^{e^{-1/\varepsilon}}\frac{\left\vert \frac {1}{(-\log x)^{a}}-\frac{1}{(-\log y)^{a}}\right\vert ^{2}}{|x-y|^{2}}dxdy=0 $$ for some $a>0$ (in case $a$ helps).