This paper analyzes the limiting behavior of L1 adaptive controllers in the presence of fast adaptation. For linear systems it is shown that the L1 adaptive controllers approximate an implementable non-adaptive linear controller, when the adaptation rate approaches infinity. This implies that both the L1 adaptive and the limiting non-adaptive control systems achieve the same control objective. The essential difference between the two systems is in the synthesis of the feedback signal. In the limiting non-adaptive system the feedback signal makes explicit use of the system inverse, while the fast estimation loop of the L1 adaptive controller approximates this inverse map, avoiding the explicit system inversion. This property of the adaptive architecture is essential for its extension to nonlinear systems and also for accommodating various hardware constraints.