Question about time limits of time-dependent perturbation theory

[ani4physics]

I have been using time-dependent perturbation theory for quite a while. Yet, one thing is still not clear to me. I have seen in many books and papers that when they calculate the transition amplitudes, they integrate from 0 to t. While in many other papers and books, the limit is taken to be - infinity to t. Those two limits give completely different results for transition amplitudes and transition probabilities. So which one to use?

 

[LostConjugate]

I would expect only 0 to t for the time that the perturbed field is switched on.

How can you use a transition amplitude based on an infinite amount of time being exposed to the field?