  Ramanujan's last letter to Hardy explored the asymptotic properties of
modular forms, as well as those of certain interesting $q$-series which he
called \emph{mock theta functions}. For his mock theta function $f(q)$, he
claimed that as $q$ approaches an even order $2k$ root of unity $\zeta$,
\[\lim_{q\to \zeta} \big(f(q) - (-1)^k (1-q)(1-q^3)(1-q^5)\cdots (1-2q + 2q^4 -
\cdots)\big) = O(1),\] and hinted at the existence of similar statements for
his other mock theta functions. Recent work of Folsom-Ono-Rhoades provides a
closed formula for the implied constant in this radial limit of $f(q)$. Here,
by different methods, we prove similar results for all of Ramanujan's 5th order
mock theta functions. Namely, we show that each 5th order mock theta function
may be related to a modular bilateral series, and exploit this connection to
obtain our results. We further explore other mock theta functions to which this
method can be applied.
