= Solution
The <Brillouin function> is odd, increases monotonically through zero, and approaches $\pm1$ for $x\to\pm\infty$. Its small-$x$ slope is $(J+1)/(3J)$. For $x\gg1$, the magnetic ions overwhelmingly occupy the largest field-aligned moment projection, so the <magnetization> reaches <magnetic saturation>:
$$
\boxed{M\longrightarrow M_s=ng\mu_BJ.}
$$
The sketch shows several $J$ values; all have the same limiting values, while their slopes differ.
\Image[/past-exam-of-the-general-relativity-and-cosmology-and-condensed-matter-physics-paper-of-part-b-of-the-physics-course-of-the-university-of-oxford/2011-brillouin-functions.png]
{title=Brillouin functions approaching magnetic saturation}
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