  In this text we study, for positive random variables, the relation between
the behaviour of the Laplace transform near infinity and the distribution near
zero. A result of De Bruijn shows that $E(e^{-\lambda X}) \sim
\exp(r\lambda^\alpha)$ for $\lambda\to\infty$ and $P(X\leq\epsilon) \sim
\exp(s/\epsilon^\beta)$ for $\epsilon\downarrow0$ are in some sense equivalent
(for $1/\alpha = 1/\beta + 1$) and gives a relation between the constants $r$
and $s$. We illustrate how this result can be used to obtain simple large
deviation results. For use in more complex situations we also give a
generalisation of De Bruijn's result to the case when the upper and lower
limits are different from each other.
