Calculating the limit with integrals of Bessel function

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I have $\alpha,\alpha_{0}$ complex numbers.

Now I would like to calculate the following:

$$\lim\limits_{\epsilon\rightarrow +0}\int\limits_{0}^{+\infty}r^{1-\epsilon}dr\int\limits_{0}^{r}r_{1}^{\epsilon-1}J_{0}(\left|\alpha r-\alpha_{0}r_{1}\right|)dr_{1}$$

where $J_{0}$ is a Bessel function. Any suggestions? However, I suspect this limit to be infinite...