For any $x \in (0,1]$, let the series $ \sum_{n=1}^{\infty}1/d_n(x)$ be the Sylvester expansion of $x$. In this paper, we consider the Hausdorff dimension of the set $$B(\alpha, \beta)= \bigg\{x \in (0,1]: \lim\limits _{n \to \infty} \frac{d_{n+1}(x)}{d^{\beta}_n(x)}=\alpha\bigg\}$$ for any $\alpha \geq 0$ and $\beta \geq 2$. As a corollary, we answer the question posed by Goldie and Smith in [6].