\displaystyle \lim_{r \to 0} r^2 (\ln r^2 - 1) = \lim_{r \to 0} \frac{\ln r^2 - 1}{\frac{1}{r^2}} \mathop{=}\limits^{\tt L'H} \lim_{r \to 0} \frac{\frac{2}{r}}{-\frac{2}{r^3}} = \lim_{r \to 0} -r^2 = 0