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The Boyle temperature is a temperature at which the second virial coefficient vanishes. At that temperature the compressibility factor has no term linear in number density, but higher terms in the virial expansion need not vanish. For a bounded strictly repulsive finite-range pair potential, $B_2(T)>0$ at every finite positive temperature, with $B_2(T)\to 0$ as $T\to \infty$; calling its Boyle temperature infinite describes this limiting cancellation, not a finite root.

Use the classical dilute-gas conventionso the second virial coefficient has units of volume. The Mayer f-function formula for an isotropic pair potential isOnly separations smaller than $a$ contribute. Define $x=\beta \epsilon >0$ and substitute $t=r^3/a^3$, so $r^{2}dr=(a^3/3)dt$:This is the second virial coefficient of a bounded cubic repulsion. For every finite positive $T$, $1-e^{-x}<x$, so $B_2(T)>0$. There is no finite positive Boyle temperature. At high temperature, expanding the exponential givesThus the intended limiting statement isAt the other extreme the classical expression tends to $2\pi a^3/3$, the hard-exclusion value for separation $a$. This is a check on the integral, not a claim that a real molecular gas stays classical at arbitrarily low temperature. A molar virial expansion would instead use $B_{2,\mathrm{m}}=N_A{B}_2$.