Reflectance of metals at low frequencies

When calculating the dielectric constant of metals using the Drude model, in the low frequency regime (infrared and beyond) one gets an approximately pure imaginary value:
$$\epsilon(\omega) \approx i\frac{4\pi n e^2\tau}{m_e\omega}$$
which gives an absorption coefficient:
$$\alpha(\omega) \approx \frac{\omega}{c}\sqrt{\frac{8\pi ne^2\tau}{m_e\omega}}$$
When looking at graphs of actual reflectivities of metals in the infrared, the reflectance is almost 100%. From this result however, I first thought that most of the incident light would be absorbed rather than reflected. Is there a physical reason for this difference, or is this a shortcoming of the Drude model?