  The C/Z_N orbifold of type II string theory has localized tachyons with m^2
ranging from -1+1/N to -2/N in units of 2/\alpha'. We show that by restricting
attention to the lightest tachyons it is possible to take a zero-slope limit
where N is taken to infinity while N\alpha' is held fixed. This is done by
applying Buscher duality in the angular direction of the cone to obtain a
supergravity solution on which the tachyons are gravitational instabilities. In
this picture, supergravity provides a natural off-shell description of the
tachyonic interactions. For example, the three-point couplings can be read off
easily (to leading order in 1/N) from the supergravity action, and are in
agreement with the on-shell couplings computed using CFT techniques.
