Answer
True
Work Step by Step
The solution for this $RL$ circuit can be expressed as:
$i(t)=c_1e^{-\frac{Rt}{L}}-\frac{V_0(Lw\cos (tw)-R\sin (tw))}{L^2w^2+R^2}\\
=c_1e^{-\frac{Rt}{L}}-\frac{V_0}{\sqrt L^2w^2+R^2}\cos(wt+tan^{-1}(\frac{R}{Lw}))$
The amplitude of the steady-state current is $\frac{V_0}{\sqrt L^2w^2+R^2}$. This decreases with increase in $w$.