  We consider mass-conserving self-similar solutions for Smoluchowski's
coagulation equation with kernel $K(\xi,\eta)= (\xi \eta)^{\lambda}$ with
$\lambda \in (0,1/2)$. It is known that such self-similar solutions $g(x)$
satisfy that $x^{-1+2\lambda} g(x)$ is bounded above and below as $x \to 0$. In
this paper we describe in detail via formal asymptotics the qualitative
behavior of a suitably rescaled function $h(x)=h_{\lambda} x^{-1+2\lambda}
g(x)$ in the limit $\lambda \to 0$. It turns out that $h \sim 1+ C
x^{\lambda/2} \cos(\sqrt{\lambda} \log x)$ as $x \to 0$. As $x$ becomes larger
$h$ develops peaks of height $1/\lambda$ that are separated by large regions
where $h$ is small. Finally, $h$ converges to zero exponentially fast as $x \to
\infty$. Our analysis is based on different approximations of a nonlocal
operator, that reduces the original equation in certain regimes to a system of
ODE.
