You guys are talking impedance which is defined only for the steady state.
Let me rephrase the question:
Does the ratio, v(t)/i(t), have any meaning or is it utter nonsense?
Assume a linear load and sinusoidal voltages and currents.
Doesn't a linear load, and sinusoidal voltage and currents, require us to be at steady state?
Besides, even if we aren't at steady state, we can take transforms of v(t) and i(t). Maybe those are V(X) and I(X). So the ratio V(X)/I(X) gives us some other function, say Z(X).
Z(X) is now a "Transfer Function" of the given circuit. And given any v(t), we can calculate V(X) and find out what I(X) would be by I(X) = V(X)* Z(X). Of course, once we know I(X), we can always transform it back and get i(t).
And Z(X) can also be transformed back to the time domain to obtain z(t), for a general frequency -f. However, if our input is not constant, it might be very difficult to transform Z(X) back to z(t,f). Maybe impossible, maybe just beyond our ability to calculate, who knows which.
So, if you have a problem with calling z(t,f) an impedence, maybe because it can vary with frequency or time, I'll provide what might be a more technically correct answer:
The transform of v(t)/i(t) is the transfer function of a particular circuit. This "transfer function" allows us to calculate the circuit response to any input. And I will stop just short of saying we can transform the "transfer function" back to find an impedence.
Steve