# I can't think of a good name

Calculus Level 5

$\displaystyle\lim _{ n\rightarrow \infty }{ \displaystyle\int _{ 0 }^{ 1 }{ \cdots \displaystyle\int _{ 0 }^{ 1 }{ f\left( \frac { n }{ \displaystyle\sum _{ k=1 }^{ n }{ \frac { 1 }{ { x }_{ k } } } } \right) g\left( \sqrt [ n ]{ \displaystyle\prod _{ k=1 }^{ n }{ { x }_{ k } } } \right) d{ x }_{ 1 }\cdots d{ x }_{ n } } } }$

Let $$f,g:[0,1]\rightarrow \mathbb{R}$$ be two continuous functions such that the limit above is equal to $Af(B)g(Ce^{-D}) \; ,$

where $$A,B,C$$ and $$D$$ are non-negative integers. Find $$A+B+C+D$$

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