  We derive analytically the asymptotic behavior of the Casimir interaction
between a sphere and a plate when the distance between them, $d$, is much
smaller than the radius of the sphere, $R$. The leading order and
next-to-leading order terms are derived from the exact formula for the Casimir
interaction energy. They are found to depend nontrivially on the dielectric
functions of the objects. As expected, the leading order term coincides with
that derived using the proximity force approximation. The result on the
next-to-leading order term complements that found by Bimonte, Emig and Kardar
[Appl. Phys. Lett. \textbf{100}, 074110 (2012)] using derivative expansion.
Numerical results are presented when the dielectric functions are given by the
plasma model or the Drude model, with the plasma frequency (for plasma and
Drude models) and relaxation frequency (for Drude model) given respectively by
9eV and 0.035eV, the conventional values used for gold metal. It is found that
if plasma model is used instead of Drude model, the error in the sum of the
first two leading terms is at most 2%, while the error in $\theta_1$, the ratio
of the next-to-leading order term divided by $d/R$ to the leading order term,
can go up to 4.5%.
