In theory, would amplitude keep increasing forever in an oscillating system in resonace? Because there is a limit in experiments modelling resonance, the amplitude of oscillation will eventually reach a limit. However, in theory, if there wouldn't be any limitations in the set up of the system, would the amplitude of oscillation keep increasing forever, without limits, at resonance?
 A: Usually, realistic modeling of a resonance includes non-zero damping, $\gamma$, in which case the amplitude does not increase to infinity:
$$
\ddot{x}+\gamma\dot{x}+\omega_0^2x=f(t).
$$
In theory without damping the amplitude may increase to infinity... but then it is just not a very good theory, since it does not reflect what happens in reality.
A: "However, in theory, if there wouldn't be any limitations in the set up of the system ..." Your "theory" wouldn't then be modelling reality very closely, because it would not be including resistive forces. If these are included, the forced oscillations, even at resonance, have a finite amplitude.
Consider an oscillatory system consisting of a body of mass $m$ subject to a restoring force $-m\omega_0^{\ 2}x$ and a resistive force $-\rho \frac{dx}{dt}$ [$\omega_0^{\ 2}$ and $\rho$ are constants]. If an oscillatory force $F=F_0\sin \omega t$ is also applied to the mass, then when transient oscillations at the natural frequency have died down to zero, we are left with oscillations of frequency $\omega/2\pi$ and of amplitude
$$x_0=\frac{F_0/m}{\sqrt{(\omega^2-\omega_0^{\ 2})^2+(\rho\omega/m )^2}}$$
When $\omega=\omega_0$, the natural frequency, we have the finite amplitude $x_0=F_0/\rho \omega$.
A: Yes. At resonance the driving force is continuously adding energy to the system. So, the amplitude will grow unbounded. This is similar to a particle undergoing constant acceleration forever. The velocity of that particle will also keep increasing unbounded.
When there is zero resistance it would not be correct to apply the theory of transients, since this theory assumes that the system will reach steady state at some point (which it doesn't).
A: Even without damping, you need to have some impulse that transfers the energy. Imagine for example a (poorly built :)) bridge oscillating in a wind; as the motion of the bridge gets faster, the wind imparts less and less energy in the resonance, until it ultimately finds and equilibrium where it's just as likely to add energy as to remove energy from the oscillation.
Or if you want an even simpler picture, imagine pushing a swing; clearly when the swing is moving just as fast as your arm providing the impulse, you're not going to transfer a whole lot of energy - and when it's moving faster, you're only going to slow it down.
