= Solution
For $q=r_0/R\to\infty$, expand the <exponential function> in powers of $1/q$:
$$
e^{-1/q}=1-\frac1q+\frac1{2q^2}-\frac1{6q^3}+\frac1{24q^4}+O(q^{-5}).
$$
The constant and linear terms cancel in $1-q+qe^{-1/q}$, giving
$$
\boxed{f(q)=\frac1{2q}-\frac1{6q^2}+\frac1{24q^3}+O(q^{-4})\sim\frac1{2q}.}
$$
For $q\to0^+$, $e^{-1/q}$ is exponentially small:
$$
\boxed{f(q)=1-q+O(qe^{-1/q})\sim1-q,\qquad f(q)\to1.}
$$
The small-$q$ correction $qe^{-1/q}$ is smaller than every positive power of $q$ beyond the displayed linear term. In dimensional terms, the large-$q$ leading <mass> is $2\pi AR^2/r_0$, while the small-$q$ limit is $4\pi AR$.
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