MPC vs LQR: when the extra machinery of MPC earns its cost

An infinite-horizon LQR and a linear MPC controller using the same quadratic stage cost and the same linear plant model can produce nearly identical control on the interior of the state space. Explain concretely what capabilities MPC adds that LQR fundamentally lacks, and identify the situations where those capabilities matter. In your answer, address: (1) constraint handling, (2) the role and limitations of the finite prediction horizon, (3) closed-loop stability and recursive feasibility (and how they can fail), and (4) the practical costs of MPC (computation, model dependence) that argue for LQR when constraints are inactive. Be specific about *why* unconstrained LQR can be recovered as a special/limiting case of linear MPC.