  We consider the Cauchy problem of the Euler-Poincar\'e equations in
$\mathbb{R}^d$ with a varying dispersion parameter $\alpha$. Based on the
convex entropy structure and the modified commutator estimates, we have proved
that the Euler-Poincar\'e equations have a uniform existence time with respect
to $\alpha$ in Sobolev spaces $H^s.$ Combined with the Bona-Simth method, we
obtain convergence of the solutions to the Euler-Poincar\'e equations as
$\alpha\to 0$ in the same space where the initial data are located.
