Energy of a Forced Oscillator

When the system is not in perfect resonance the oscillator will sometimes gain energy and sometimes lose energy. Only at resonance are the conditions such that the oscillator will always extract energy from the driver. When you drive the oscillator for a while it reaches steady state. At that point the oscillator is saturated and can't absorb any more energy. This plot shows the steady amplitude for damping (black lines) or no damping (red line):

At resonance ($omega=omega_c$) and zero damping the oscillator will always gain energy so in principle if you waited long enough you could get arbitrarily high energy. The steady state amplitude is infinity. Having zero damping is of course not really realistic.

Source of graph: https://en.wikipedia.org/wiki/Resonance

Transmissability means output/input.

Edit in response to comment:

In this plot I drew the driving force $F_0sinomega t$ on top of the velocity $v(t)=x'(t)$ of the steady state solution. The driving force is blue and $v(t)$ is orange. Depending on whether the driving force is above/below/on resonance you get a different phase offset. When the driving force is too slow ($omega<omega_0$) or too fast ($omega>omega_0$) the velocity is almost 90$^circ$ out of phase. This means about half the time the driving force adds energy (when the driving force and velocity have the same sign) and about half the time the driving force removes energy (they have the opposite sign). As $omega$ goes near $omega_0$ they become in phase and the driving always adds energy. Of course the damping always removes energy from the system.