= Solution
For $x=h\nu/(k_BT)\ll1$, the <Taylor expansion> $e^x-1=x+O(x^2)$ gives the <Rayleigh-Jeans law> for equilibrium radiation,
$$
\boxed{U_\nu\simeq\frac{8\pi k_BT\nu^2}{c^3}.}
$$
This is the classical limit: each electromagnetic mode has mean thermal <energy> approximately $k_BT$. Extending it to arbitrarily large <frequencies> would give a divergent integrated <energy density>, the <ultraviolet catastrophe>.
For $x\gg1$, $e^x-1\simeq e^x$, giving the <Wien approximation>
$$
\boxed{U_\nu\simeq\frac{8\pi h\nu^3}{c^3}e^{-h\nu/(k_BT)}.}
$$
<Planck's quantum hypothesis> assigns equally spaced mode energies $E_n=nh\nu$. When the spacing is small relative to $k_BT$, many levels are occupied and the classical approximation works. When $h\nu\gg k_BT$, exciting even one quantum is exponentially unlikely, suppressing the high-frequency tail. The single expression $\langle E\rangle=h\nu/(e^x-1)$ contains both limits. Comparing <temperature> directly to $h\nu$ would be dimensionally incorrect; the low-temperature condition means $k_BT\ll h\nu$.
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