  We consider a matrix pencil whose coefficients depend on a positive parameter
$\epsilon$, and have asymptotic equivalents of the form $a\epsilon^A$ when
$\epsilon$ goes to zero, where the leading coefficient $a$ is complex, and the
leading exponent $A$ is real. We show that the asymptotic equivalent of every
eigenvalue of the pencil can be determined generically from the asymptotic
equivalents of the coefficients of the pencil. The generic leading exponents of
the eigenvalues are the "eigenvalues" of a min-plus matrix pencil. The leading
coefficients of the eigenvalues are the eigenvalues of auxiliary matrix
pencils, constructed from certain optimal assignment problems.
