For a driven damped oscillation, if the driven force $F=F_{ 0 }cos(\omega t )$, then the solution to the motion is $$x=Acos(\omega t+\varphi )$$.

• Why the oscillation and the driven force must have the same frequency but can be out of phase?

The solution to $\varphi$ is $tan(\varphi )=\frac { \omega \gamma }{ { { \omega }_{ 0 } }^{ 2 }-{ \omega }^{ 2 } }$.

• Why would $\varphi$ goes to zero(F and x in phase) if $\omega$ goes to zero?