  The 3D-index of Dimofte-Gaiotto-Gukov is an interesting collection of
$q$-series with integer coefficients parametrised by a pair of integers and
associated to a 3-manifold with torus boundary. In this note, we explain the
structure of the asymptotic expansions of the 3D-index when $q=e^{2\pi i\tau}$
and $\tau$ tends to zero (to all orders and with exponentially small terms
included), and discover two phenomena: (a) when $\tau$ tends to zero on a ray
near the positive real axis, the horizontal asymptotics of the meromorphic
3D-index match to all orders with the asymptotics of the Turaev-Viro invariant
of a knot, in particular explaining the Volume Conjecture of Chen-Yang from
first principles, (b) when $\tau \to 0$ on the positive imaginary axis, the
vertical asymptotics of the 3D-index involves periods of a plane curve (the
$A$-polynomial), as opposed to algebraic numbers, explaining some predictions
of Hodgson-Kricker-Siejakowski and leading to conjectural identities between
periods of the $A$-polynomial of a knot and integrals of the Euler
beta-function.
