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Differentiate $I(\epsilon):=2\int_0^\infty e^{-\epsilon x^2}(1+x^2)^{-1/2}dx$ with respect to $\epsilon$ under the sign of integral; change variable putting $u:=\epsilon x^2$. We get $$I'(\epsilon) = -\frac{1}{\epsilon} \int_0^\infty e^{-u}\sqrt{\frac{u}{u+\epsilon}}du= -\frac{1}{\epsilon}\big(1+o(1)\big)\\ ,$$ by the dominated convergence theorem, and integrating $$I(\epsilon)=-\log(\epsilon)\big(1+o(1)\big),\\ \\ \mathrm{as }\\ \epsilon\to0\\ .$$