I can't speak for other nations, but they still teach numeric analysis in Chinese universities as an undergraduate course. In my university it is a required subject. Many of us have countless dreadful memories of Runge-Kutta method, Euler's method, Newton's method, rate of convergence, numerical stability and error margins, just to name a few of the dreads...
Yes, perturbation methods are still taught and are still recognized as important. The first math course I took during my graduate degree (Intro analytic methods) covered it, for example.
