This study initiates a resource-aware theory of language generation in the limit under space efficiency constraints. A learner observes an adversarial positive stream from a target language K and must output a hallucination-free hypothesis L while omitting at most Δ strings. The research focuses on DFAs with s states over an alphabet of size k as the hypothesis class for memory-bounded learners. In the exponential-space regime, the authors prove that a learner can exactly identify the target language K. Under stricter memory budgets, they present a streaming algorithm using poly(s,k) space that converges to a hypothesis with a generation gap of Δ= O(k^{2s-2}). This learned hypothesis captures every string in K of length at least 2s-1. The results are complemented by a near-matching lower bound derived from communication complexity, showing that achieving Δ≤ k^{(1-ε)s} requires k^{Ω(εs)} memory. These findings reveal a sharp transition between polynomial-space generation and exponential-space exact identification.