MHB A generalization of the limit definining \$e\$.

[Alone]

I was thinking of generalizing the limit of $\lim_{n\to \infty} (1+x/n)^n=\exp(x)$. What do we know of $$\lim_{n_1\to \infty , n_2 \to \infty , \ldots , n_k \to \infty } (1+\prod_{i=1}^k x_i/n_i)^{\prod_{i=1}^k n_i}$$?

 

[Alone]

Well, now I think it's trivial if we define $m=\prod_i n_i$, then the limit should be: $\exp(\prod_i x_i)$, nothing new here.

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How about instead of multiplication we have addition, i.e.:
$\lim (1+\sum_{i=1}^k x_i/n_i)^{\prod_{i=1}^k n_i}$, which seems tougher to find.
How would you go about solving this limit?

Thanks!