  In this paper, we justify the low Mach number limit of the steady
irrotational Euler flows for the airfoil problem, which is the first result for
the low Mach number limit of the steady Euler flows in an exterior domain. The
uniform estimates on the compressibility parameter $\varepsilon$, which is
singular for the flows, are established via a variational approach based on the
compressible-incompressible difference functions. The limit is on the
H\"{o}lder space and is unique. Moreover, the convergence rate is of order
$\varepsilon^2$. It is noticeable that, due to the feature of the airfoil
problem, the extra force dominates the asymptotic decay rate of the
compressible flow to the infinity. And the effect of extra force vanishes in
the limiting process from compressible flows to the incompressible ones, as the
Mach number goes to zero.
