Considering the Cauchy problem for the modified finite-depth-fluid equation $partial_tu-G_delta(partial_x^2u)mp u^2u_x=0, u(0)=u_0$, where $G_delta f=-i ft ^{-1}[coth(2pi delta xi)-frac{1}{2pi delta xi}]ft f$, $deltages 1$, and $u$ is a real-valued function, we show that it is uniformly globally well-posed if $u_0 in H^s (sgeq 1/2)$ with $ orm{u_0}_{L^2}$ sufficiently small for all $delta ges 1$. Our result is sharp in the sense that the solution map fails to be $C^3$ in $H^s (s<1/2)$. Moreover, we prove that for any $T>0$, its solution converges in $C([0,T]; H^s)$ to that of the modified Benjamin-Ono equation if $delta$ tends to $+infty$.
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