  In this article, for a certain subset $\mathcal{X}$ of the extended set of
rational numbers, we introduce the notion of {\it best
$\mathcal{X}$-approximations} of a real number. The notion of best
$\mathcal{X}$-approximation is analogous to that of best rational
approximation. We explore these approximations with the help of
$\mathcal{F}_{p^l}$-continued fractions, where $p$ is a prime and
$l\in\mathbb{N}$, we show that the convergents of the
$\mathcal{F}_{p^l}$-continued fraction expansion of a real number $x$
satisfying certain maximal conditions are exactly the best
$\mathcal{F}_{p^l}$-approximations of $x$.
