differential equations – How to implement limit boundary condition in solving PDE

I have to solve a partial differential equation for a function $F(x,t)$ where one of the boundary condition is formulated in terms of a limit:

$lim_{xrightarrow +infty} e^x partial_xF(x,t)=0$

Is it possible to implement this as a boundary condition in NDSolve (or possibly NDSolve`FiniteDifferenceDerivative)?