With the term "productory" I just mean
$\Pi_{i=m}^nx_i$ but I do not know the english term. My question is:
is there a limit for such an expression in the same sense as the limit of a sum is an integral, i.e. $\lim_{\delta x\rightarrow 0}\Sigma y(x)\delta x = \int y(x) dx$?
Has it any application?
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Yes, there is!
It is called product integral. I think it was introduced by Vito Volterra for probability applications. A standard reference about it is: "Product Integration with Applications to Differential Equations", by Dollard and Friedman.
Also, check out this question here.
