Let \(q_n(x)\) be the denominator of the \(n\)-th convergent of the continued fraction expansion \([a_1(x),a_2(x),\dots,a_n(x),\dots]\) of an irrational number \(x\in(0,1)\). \(q_n(x)\) satisfies the recursive formula \(q_n(x)=a_n(x)q_{n-1}(x)+q_{n-2}(x)\) (\(n\ge 1\)) with \(q_{-1}(x)=0\) and \(q_0(x)=1\). In this paper, the Baire category of the set \[ E(\alpha,\beta):=\left\{x\in(0,1)\backslash\mathbb Q:\liminf_{n\to\infty}\frac{\log a_{n+1}(x)}{\log q_n(x)}=\alpha,\\ \limsup_{n\to\infty}\frac{\log a_{n+1}(x)}{\log q_n(x)}=\beta\right\} \] for all \(0\le\alpha\le\beta\le\infty\) is studied. A set is said to be of first category if it can be represented as a countable union of nowhere dense sets. A set is residual if its complement is of first category. It is proved that the set \(E(\alpha,\beta)\) is residual if and only if \(\alpha=0\) and \(\beta=\infty\).