  We prove exact leading-order asymptotic behaviour at the origin for
nontrivial solutions of two families of nonlocal equations. The equations
investigated include those satisfied by the cusped highest steady waves for
both the uni- and bidirectional Whitham equations. The problem is therefore
analogous to that of capturing the $120\unicode{xb0}$ interior angle at the
crests of classical Stokes' waves of greatest height. In particular, our
results partially settle conjectures for such extreme waves posed in a series
of recent papers. Our methods may be generalised to solutions of other nonlocal
equations, and can moreover be used to determine asymptotic behaviour of their
derivatives to any order.
