Transients in series RL circuit with AC source
The &quot;transient&quot; response dies off simply because that is the definition of transient in this context. The transient response is the result of an event at a specific point in time, such as opening/closing a switch or turning on a source. This is in contrast to the steady-state response, which assumes that the stimulus is a periodic (usually sinusoidal) waveform that has existed forever and will continue forever.
You can have both a transient and a steady-state response if, as you indicate, you suddenly connect a source of any kind.