Consider the canonical and grand canonical partition functions given by $$Z_C=\sum\limits_{i}g(E_i)e^{-\beta E_i}$$ and $$Z_G=\sum\limits_{i}g(E_i)e^{-\beta (E_i-\mu)}$$ respectively with $\beta=\frac{1}{k_BT}$.
Questions
$\bullet$ What happens to these partition functions in the limit $\beta\to\infty$? Does it become a constant (in the sense that independent of $E_i$)?
$\bullet$ What is the physical significance of the limiting result (whatever it turns out)?
Update: The existing answer doesn't include the role of $g(E)$ i.e., the degeneracy of the energy level $E$ which is crucial for taking the limit. It also doesn't mention what happens to the grand partition function in the same limit. It is trickier because $\mu$ itself changes with temperature $T$.