Mentioned in the wikipedia article, the $0$th power mean is defined to be the geometric mean. Why is this? I understand that a convenient consequence is that the means are ordered by their exponent. But is there an intuitive reason why the $0$th mean should be the geometric mean? Is it true that
$$\lim_{r\downarrow 0} \left(\frac{a_1^r + \dotsb + a_n^r}{n}\right)^{1/r}= \left( a_1 \dotsb a_n\right)^{1/n}?$$
If I take the logarithm and use L'Hospital's rule I get
$$\ln L = \lim_{r\downarrow 0} \frac{a_1^r + \dotsb + a_n^r}{n\left( a_1^r \ln{a_1} + \dotsb + a_n^r \ln{a_n} \right)}.$$
How might we evaluate this?