  Let $f(n)$ be an arithmetic function with $f(1)\neq0$ and let $f^{-1}(n)$ be
its reciprocal with respect to the Dirichlet convolution. We study the
asymptotic behaviour of $|f^{-1}(n)|$ with regard to the asymptotic behaviour
of $|f(n)|$ assuming that the latter one grows or decays with at most
polynomial or exponential speed. As a by-product, we obtain simple but
constructive upper bounds for the number of ordered factorizations of $n$ into
$k$ factors.
