  We consider a particular large-radius limit of the worldsheet $S$-matrix for
strings propagating on $AdS_5 \times S^5$. This limiting theory interpolates
smoothly between the so-called plane-wave and giant-magnon regimes of the
theory. The sigma model in this region simplifies; it stands as a toy model of
the full theory, and may be easier to solve directly. The $S$ matrix of the
limiting theory is non-trivial, and receives contributions to all orders in the
$\alpha'$ expansion. We analyze a guess for the full worldsheet $S$ matrix that
was formulated recently by Beisert, Hernandez and Lopez, and Beisert, Eden, and
Staudacher, and take the corresponding limit. After doing a Borel resummation
we find that the proposed $S$ matrix reproduces the expected results in the
giant-magnon region. In addition, we rely on general considerations to draw
some basic conclusions about the analytic structure of the $S$ matrix.
