We give a new characterization of the hyperarithmetic sets: a set $X$ of integers is recursive in $e_\alpha$ if and only if there is a Turing machine which computes $X$ and "halts" in less than or equal to the ordinal number $\omega^\alpha$ of steps. This result represents a generalization of the well-known "limit lemma" due to J. R. Shoenfield [Sho-1] and later independently by H. Putnam [Pu] and independently by E. M. Gold [Go]. As an application of this result, we give a recursion theoretic analysis of clopen determinacy: there is a correlation given between the height $\alpha$ of a well-founded tree corresponding to a clopen game $A \subseteq \omega^\omega$ and the Turing degree of a winning strategy $f$ for one of the players--roughly, $f$ can be chosen to be recursive in $0^\alpha$ and this is the best possible (see $\S 4$ for precise results).