= Solution
For $g>0$, $B\to+\infty$ at fixed $T>0$ selects $m=-J$, so all moments align with the field:
$$
\boxed{M\longrightarrow Ng\mu_BJ.}
$$
This is <magnetic saturation>. The <Curie response of an arbitrary independent spin> follows at weak field: $\coth z=z^{-1}+z/3+O(z^3)$ gives
$$
\boxed{M=\frac{N(g\mu_B)^2J(J+1)}{3k_BT}B+O(B^3)\longrightarrow0.}
$$
This is the <Curie law> of independent moments with <paramagnetism>. Thermal populations are symmetric at zero field, so the model has no spontaneous <magnetization>. Both limits hold at fixed positive <temperature>; taking $T\to0$ first is a different limiting procedure.
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