For a strictly proper transfer function with no poles/zeros at infinity, I understand that it gives G(inf) = 0. What does it for a proper TF?
For a transfer function with more finite poles than finite zeros, the gain goes to zero as the frequency increases without bound.
For a transfer function with an equal number of finite poles and finite zeros, the gain will be non-zero as the frequency increases without bound. This is given by the "D" term in the state space representation of the transfer function.