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In this article, for a certain subset $\mathcal{X}$ of the extended set ofrational 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 rationalapproximation. 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 $x$.

Best Approximations by $\mathcal{F}_{p^l}$-Continued Fractions