In the canonical ensemble, the total energy is not conserved.
( 
). What are the fluctuations in the
energy? The energy fluctuations are given by the root mean
square deviation of the Hamiltonian from its average 
:
Therefore
But
Thus,
Therefore, the relative energy fluctuation 
is given by
Now consider what happens when the system is taken to be
very large. In fact, we will define a formal limit
called the thermodynamic limit, in which
and 
such that
N/V remains constant.
Since 
and E are both extensive variables, 
and
,
But would be exactly 0 in the microcanonical ensemble.
Thus, in the thermodynamic limit, the canonical and microcanonical ensembles
are equivalent, since the energy fluctuations become vanishingly small.