Quantum information processing comprises physical processes, which obey the quantum speed limit (QSL): high speed requires strong driving. Single-qubit gates using Rabi oscillation, which is based on the rotating wave approximation (RWA), satisfy this bound in the form that the gate time (T) is inversely proportional to the Rabi frequency (Ω), characterizing the driving strength. However, if the gate time is comparable or shorter than the qubit period (T_{0} \equiv 2\pi / \omega_{0}), the RWA actually breaks down since the Rabi frequency has to be large compared to the qubit frequency (\omega_{0}) due to the QSL, which is given as (T \gtrsim \pi/Ω). We show that it is possible to construct a universal set of single-qubit gates at this strong-coupling and ultrafast regime, by adjusting the central frequency (\omega) and the Rabi frequency (Ω) of the driving pulse. We observe a transition in the scaling behavior of the central frequency from the long-gate time regime ((T \gg T_{0})) to the short-gate time ((T \ll T_{0})) regime. In the former, the central frequency is nearly resonant to the qubit, i.e., (\omega \simeq \omega_{0}), whereas in the latter, the central frequency is inversely proportional to the gate time, i.e., (\omega \sim \pi/T). We identify the transition gate time at which the scaling exponent (n) of the optimal central frequency (\omega \sim T^{n}) changes from (n=0) to (n=-1).