We study the real numbers with partial quotients diverging to infinity in a subsequence. We show that if the subsequence has positive density then such sets have Hausdorff dimension equal to 1/2. This generalizes one of the results obtained in [C. Y. Cao, B. W. Wang and J. Wu, The growth speed of digits in infinite iterated function systems, Studia. Math. 217(2) (2013) 139-158; I. J. Good, The fractional dimensional theory of continued fractions, Proc. Cambridge Philos. Soc. 37 (1941) 199-228].