  Let $\left\{ Z_t \right\}_t$ be a one-parameter family of complex toric
hypersurfaces of dimension $d \geq 1$. We compute asymptotics of period
integrals for $\left\{ Z_t \right\}_t$ by applying the method of
Abouzaid--Ganatra--Iritani--Sheridan, which uses tropical geometry. As
integrands, we consider Poincar\'{e} residues of meromorphic $(d+1)$-forms on
the ambient toric variety, which have poles along the hypersurface $Z_t$. The
cycles over which we integrate them are spheres and tori which correspond to
tropical $(0, d)$-cycles and $(d, 0)$-cycles on the tropicalization of $\left\{
Z_t \right\}_t$ respectively. In the case of $d=1$, we explicitly write down
the polarized logarithmic Hodge structure of Kato--Usui at the limit as a
corollary.
