\[ \large \displaystyle\lim _{ x\rightarrow \infty }{ \left( (f(x+2)\Gamma (x+2))^{ 1/(x+1) }-(f(x+1)\Gamma (x+1))^{ 1/x } \right) } =A{ e }^{ -B } \]

Let \(f\) be a polynomial whose coefficient in the leading term is positive and it satisfy the equation above for positive integers \(A\) and \(B\). Find \(A+B\).

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