1. Permittivity is the ability of a material to resist the formation of electric fields inside it. Hence, a metal has infinite permittivity. 

2. But, doesn't permittivity relate to Polarization i.e. the ability to form dipoles within it? The higher the permittivity, a material tends to become a dielectric. But, conductivity of metals overshadows its dipole formation ability. Since a metal has zero polarizability (P = ε0ΧeE), Χe=0. Χe= εr-1=0, εr =1. By this, doesn't a metal have permittivity equal to one?

3. If one keeps a metal between capacitor plates, its capacitance equals 0? Thus, its relative permittivity equals zero (εr=Cx/C0)?