I'm trying to prove that
$\displaystyle \int_{[2,\infty]} \frac{x\,dx}{\sqrt{(x^2-\epsilon^2)(x^2-1)(x-2)}}\;\;$ converges to $\;\;\displaystyle \int_{[2,\infty]}\frac{dx}{\sqrt{(x^2-1)(x-2)}}\;\;$ as $\epsilon\rightarrow\,0$
I thought I could use Dominated Convergence Theorem, but I'm having trouble finding an integrable dominating function because of what happens at x=2. Can I use the 'almost everywhere' part to split the interval somehow?
Is the theorem applicable, or should I be thinking about this differently?