This paper studies the problem of learning computable functions in the limit by extending Gold's inductive inference framework to incorporate computational observations and restricted input sources. Complimentary to the traditional Input-Output Observations, we introduce Time-Bound Observations, and Policy-Trajectory Observations to study the learnability of general recursive functions under more realistic constraints. While input-output observations do not suffice for learning the class of general recursive functions in the limit, we overcome this learning barrier by imposing computational complexity constraints or supplementing with approximate time-bound observations. Further, we build a formal framework around observations of computational agents and show that learning computable functions from policy trajectories reduces to learning rational functions from input and output, thereby revealing interesting connections to finite-state transducer inference. On the negative side, we show that computable or polynomial-mass characteristic sets cannot exist for the class of linear-time computable functions even for policy-trajectory observations.