I am reading the Quantum Mechanics, 2nd edition by Bransden and Joachain. On page 777, the book gives an example of Dirac delta function.

$\delta_\epsilon (x) = \frac{\epsilon}{\pi(x^2 + \epsilon^2)}$

I am wondering how I can show $\lim_{x\to 0+} \int_{a}^{b} \delta_\epsilon (x) dx$ equals to 1. I've thought about using L'Hospital's rule, but there are two variables, $x$ and $\epsilon$, so seems I cannot use the rule directly. I've been stuck in this for some time. Will be grateful if some one can point the direction for me. Thanks.