Isn't the output adjustment always one step behind the change in the
input? Is there an input frequency or dVin/dt at which the op-amp
would not be able to stabilize the output frequency?
Yes it is and it boils down to a criteria called phase margin - this is a measure of the op-amps stability across the range of frequencies it is useful for.
Taken to extremes, the op-amp could be slow enough so that any counter measure taken by it to cancel an unwanted alteration in the output might actually produce positive feedback. This is how an oscillator works and most of us have heard of output instabilities in op-amps now and then.
So phase margin is the key to stability. If you look at the op-amp open-loop response below you might see what I mean: -
You can see that for low to mid frequencies the open-loop phase shift on the output is about 90 degrees - this is typical of most op-amps ran with open-loop gain and, of course, when you close the loop, the phase shift is 180 degrees. However, that isn't the case open- loop because an open-loop op-amp is like an integrator.
However, as frequency rises from the low/mid range, the "delay" factor starts to get involved and this is the same as shifting the phase. There is a point reached where the open-loop phase shift would naturally stray from 90 degrees and reach 0 degrees AND, if the op-amp is still capable of producing a gain greater than unity then there is a potential problem when the loop is closed.
Let's, for the sake of argument say that the open-loop gain were ten at a phase shift of zero degrees and, let's also say we wanted the closed-loop op-amp to have unity gain. So now, if we analyze the op-amp (open loop) AND the feedback network, the net gain back to the input (at the frequency that causes zero degrees phase shift) is over ten and bingo, the circuit oscillates.
In the picture above, we have a decent op-amp that can be run with a closed-loop gain of unity (worst case consideration) because the phase margin is approximately 45 degrees i.e. the phase shift is 45 degrees away from it becoming an oscillator when the gain falls to unity.
However, like any control loop, the down side to being able to remain stable is that fast output fluctuations (whether power supply induced or load induced or demand induced) cannot be sufficiently dealt with as they can when the fluctuations are a lower frequency i.e. the control-loop runs out of steam so to speak.