We prove that in dimensions three and higher the Landau-Lifshitz-Gilbert equation with small initial data in the critical Besov space is globally wellposed in a uniform way with respect to the Gilbert damping parameter. Then we show that the global solution converges to that of the Schrödinger maps in the natural space as the Gilbert damping term vanishes. The proof is based on some studies on the derivative Ginzburg-Landau equations.2010 Mathematics Subject Classification. 35Q55.