  Let $(\xi_1, \eta_1)$, $(\xi_2, \eta_2),\ldots$ be independent identically
distributed $\mathbb{R}^2$-valued random vectors. Assuming that $\xi_1$ has
zero mean and finite variance and imposing three distinct groups of assumptions
on the distribution of $\eta_1$ we prove three functional limit theorems for
the logarithm of convergent discounted perpetuities $\sum_{k\geq
0}e^{\xi_1+\ldots+\xi_k-ak}\eta_{k+1}$ as $a\to 0+$. Also, we prove a law of
the iterated logarithm which corresponds to one of the aforementioned
functional limit theorems. The present paper continues a line of research
initiated in the paper Iksanov, Nikitin and Samoillenko (2022), which focused
on limit theorems for a different type of convergent discounted perpetuities.
