  We provide (for both the real and complex settings) a family of constants, $%
(C_{m})_{m\in \mathbb{N}}$, enjoying the Bohnenblust--Hille inequality and such
that $\displaystyle\lim_{m\rightarrow \infty}\frac{C_{m}}{C_{m-1}}=1$, i.e.,
their asymptotic growth is the best possible. As a consequence, we also show
that the optimal constants, $(K_{m})_{m\in \mathbb{N}}$, in the
Bohnenblust--Hille inequality have the best possible asymptotic behavior.
Besides its intrinsic mathematical interest and potential applications to
different areas, the importance of this result also lies in the fact that all
previous estimates and related results for the last 80 years (such as, for
instance, the multilinear version of the famous Grothendieck Theorem for
absolutely summing operators) always present constants $C_{m}$'s growing at an
exponential rate of certain power of $m$.
