How do I put low temperature limit in this problem?

We are given the following problem as an assignment:

Consider a system of $N$ non-interacting electrons /cm$^2$. Each can occupy either a bound state with energy $epsilon=-E_b$ or a free particle continuum with $epsilon=p^2/2m$. The number of free electrons (i.e., electrons in the continuum) as a function of T in the low-temperature limit is

• $NN_ce^{-E_b/k_BT}$
• $sqrt{NN_c}e^{-E_b/2k_BT}$
• $sqrt{NN_c}e^{-E_b/k_BT}$
• $sqrt{N/N_c}e^{-E_b/2k_BT}$

where $N_c=2(2pi mk_BT/h^2)^{3/2}$.

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The density of state for the free electrons can be found as
$$g(k)dk=2frac{4pi k^2dk}{(2pi/L)^3}$$

$$g(epsilon)depsilon=frac{4pi }{h^3}(2m)^{3/2}sqrt{epsilon} depsilon$$
The single electron partition function
$$Z_1=int g(epsilon)e^{-beta epsilon}depsilon+e^{beta E_b}$$

$$Z_1=VN_c+e^{beta E_b}$$
The probability for electron to be in continuum then
$$P=frac{VN_c}{VN_c+e^{beta E_b}}$$
The joint density
$$P=left(frac{VN_c}{VN_c+e^{beta E_b}}right)^N$$
The number of free electrons
$$n= Nleft(frac{VN_c}{VN_c+e^{beta E_b}}right)^N$$
I don’t if what I have done is right or wrong? But it seems correct to me. How do I put a low-temperature limit? I know, I have to take $beta rightarrow infty$. But I don’t know how to do that? Can any help me with this?