For time-delay systems, the asymptotic behavior analysis of the critical imaginary roots w.r.t. the infinitely many critical delays is an open problem. In order to find a general solution, we will exploit the link between the asymptotic behavior of critical imaginary roots and the asymptotic behavior of frequency-sweeping curves, from a new analytic curve perspective. As a consequence, we will establish a frequency-sweeping framework with three main results: (1) A finer (regularity-singularity) classification for time-delay systems will be obtained. (2) The general invariance property will be proved and hence the asymptotic behavior of the critical imaginary roots w.r.t. the infinitely many critical delays can be adequately studied. (3) The complete stability problem can be fully solved. Moreover, the frequency-sweeping framework is extended to cover a broader class of time-delay systems. Finally, the geometric insights of frequency-sweeping curves are investigated. Consequently, some deeper properties on the asymptotic behavior of time-delay systems and the link to frequency-sweeping curves are found.