What is the meaning of current at infinite time for a series RLC circuit formula
If you have the solution to the differential equation, then it's simple, allow t to go to infinity and the exponential term goes to zero. 

The function you show is underdamped (complex roots), overdamped will have real roots and critically damped has repeated real roots. 

If you don't have the solution, well you know that a capacitor acts as an open circuit after a long time, and an inductor acts as a short. 

So a parallel RLC circuit will have zero voltage after a long time, and a series RLC circuit will have zero current. Regardless of the (presumed finite) initial conditions.