This problem is from Galaxies in the Universe: An Introduction, 2nd Edition
For the first question, it is solved here
But for the second question, it seems wierd that:
$$ \lim_{R\to 0}\Sigma(R) = \int_0^\infty 2n_0 (\frac{r_0}{r})^{\alpha} \mathrm{d}r = \int_0^b 2n_0 (\frac{r_0}{r})^{\alpha} \mathrm{d}r + \int_b^\infty 2n_0 (\frac{r_0}{r})^{\alpha} \mathrm{d}r $$
The first term is converged when $a<1$ but the second term is converged when $a>1$
Maybe I should not use the $\int_0^\infty 2n_0 (\frac{r_0}{r})^{\alpha} \mathrm{d}r$ and get back to:
$$ \lim_{R\to 0}\int_R^\infty 2n_0 (\frac{r_0}{r})^{\alpha} \frac{r\mathrm{d}r}{\sqrt{r^2-R^2}} $$