What is the significance of heat capacity at extreme temperatures?

[aaaa202]

So heat capacity is the change in the mean value of the internal energy when the temperature is changed: C = d<E>/dt
But I would like a little more intuition than that. T^(-1) = dS/d<E>, so going back to the intuition that the inverse of the temperature is a measure of how disordered the system becomes when we change the energy, what does this tell us that heat capacity is a measure for?
The reason I am asking is I got an exercise, where the heat capacity goes from 0 to a max and then back to sorry for T=[0,∞). I am asked to interpret the meaning that C=0 for T=0 or T=∞, but I can't get the right intuition in terms of microstates etc.

 

[daveyrocket]

Look for an equation where you are expressing C in terms of S and T rather than anything to do with E. (Hint: Multiply your two equations together)

 

[aaaa202]

well all I could get from that is:

dU = TdS
dU=CdT
so
C=T dS/dT which is just the chain rule for dU/dT=C

Was this what you were looking for? This doesn't for me give a lot of new insight.

 

[aaaa202]

I could also ask another way. Why is it intuitively that solid materials have C=0 at T=0?