<p>In this article, we study the limit behavior of solutions to an energy-critical complex Ginzburg–Landau equation. Via energy method, we establish a rigorous theory of the zero-dispersion limit from energy-critical complex Ginzburg–Landau equation to energy-critical nonlinear heat equation in dimensions three and four for both the defocusing and focusing cases. Furthermore, we derive the inviscid limit of energy-critical complex Ginzburg–Landau equation from energy-critical nonlinear Schrödinger equation in dimension four for the focusing case.</p>