My professor in class used the below result to generalise the concept of averages-
$$\lim_{n\rightarrow0}(\sum_{j=1}^{m}w_ja_j^n)^{1/n}=\prod_{j=1}^ma_j^{w_j}$$ $$\sum_{j=1}^{m}w_j=1$$
All the constants above are strictly positive. He used the above the expression with n as the parameter and mentioned that for its different values, we'll get the different means. e.g. $n=1$ is clearly the weighted arithmetic mean, $n\rightarrow\infty$ would yield the maximum of the series $a_j$ and so on. These other results make sense, but I'm unable to wrap my head around the weighted geometric average.
Is there a way to prove this? Any directions to what this result is called or a reference paper would be appreciated.
Note that $\sum_{j=1}^mw_ja_j^n\xrightarrow{n\to 0}1$, hence by taking logarithms and recalling $\log(x)\sim x-1$ as $x\to 1$ we have \begin{align} \log\left(\sum_{j=1}^mw_ja_j^n\right)^{1/n} &=\frac 1n\log\left(\sum_{j=1}^mw_ja_j^n\right)\\ &\sim\frac 1n\left(\sum_{j=1}^mw_ja_j^n-1\right)\\ &=\frac 1n\left(\sum_{j=1}^mw_ja_j^n-\sum_{j=1}^mw_j\right)\\ &=\sum_{j=1}^mw_j\frac{a_j^n-1}n\\ &=\sum_{j=1}^mw_j\frac{e^{n\log(a_j)}-1}n\\ &\to\sum_{j=1}^mw_j\log(a_j)\\ &=\log\left(\prod_{j=1}^ma_j^{w_j}\right) \end{align} as $n\to 0$.