  We study the limit behavior of the solutions to energy-critical complex
Ginzburg-Landau equation. We give a rigorous theory of the zero-dispersion
limit from energy-critical complex Ginzburg-Landau equation to energy-critical
nonlinear heat equation for dimensions 3 and 4 in both the defocusing and
focusing cases by energy method. Furthermore, we also show the invisicid limit
of energy-critical complex Ginzburg-Landau equation to energy-critical
nonlinear Schr\"odinger equation for dimension 4 in the focusing case.
