  Very often traditional approaches studying dynamics of self-similarity
processes are not able to give their quantitative characteristics at infinity
and, as a consequence, use limits to overcome this difficulty. For example, it
is well know that the limit area of Sierpinski's carpet and volume of Menger's
sponge are equal to zero. It is shown in this paper that recently introduced
infinite and infinitesimal numbers allow us to use exact expressions instead of
limits and to calculate exact infinitesimal values of areas and volumes at
various points at infinity even if the chosen moment of the observation is
infinitely faraway on the time axis from the starting point. It is interesting
that traditional results that can be obtained without the usage of infinite and
infinitesimal numbers can be produced just as finite approximations of the new
ones.
