= Resonance and high-frequency asymptote
To find the displacement peak, minimize the squared denominator:
$$
\frac{d}{d\omega}[(k-m\omega^2)^2+b^2\omega^2]
=2\omega[-2m(k-m\omega^2)+b^2].
$$
The nonzero minimum gives
$$
\boxed{\omega_a=\sqrt{\frac km-\frac{b^2}{2m^2}}},\qquad b^2<2mk.
$$
For stronger damping there is no positive-frequency displacement maximum. When $\omega\gg\omega_0$ and $b\ll m\omega$,
$$
\boxed{|R(\omega)|\simeq\frac1{m\omega^2}},\qquad R\simeq-\frac1{m\omega^2}.
$$
The inertia then dominates the restoring and damping <forces>.
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