  Let k be a field of characteristic 0, R = k[x_1, ..., x_d] be a polynomial
ring, and m its maximal homogeneous ideal. Let I be a homogeneous ideal in R.
In this paper we investigate asymptotic behaviour of the quotient between the
length of local cohomology group H^0_m(R/I^n) and n^d. We show that this
quantity always has a limit as n goes to infinity. We also give an example for
which the limit is irrational; in particular, this proves that the length of
H^0_m(R/I^n) is not asymptotically a polynomial in n.
