In this paper, we address the problem of energy efficient localized routing in wireless ad hoc networks. Numerous energy aware routing protocols were proposed to seek the power efficiency of routes. Among them, several geographical localized routing protocols were proposed to help making smarter routing decision using only local information and reduce the routing overhead. However, most of the proposed localized routing methods cannot theoretically guarantee the power efficiency of their routes. In this paper, we give the first localized routing algorithm, called Localized Energy Aware Restricted Neighborhood routing (LEARN), which can guarantee the power efficiency of its route asymptotically almost sure. Given destination node t, an intermediate node u will only select a certain neighbor v such that ∠ vut ≤ a for a parameter α < π/3 in our LEARN method. We theoretically prove that for a network, formed by nodes that are produced by a Poisson distribution with rate n over a compact and convex region Ω with unit area, when the transmission range rn = √ LEARN routing protocol will find the route for any pair of nodes asymptotically almost sure. When the transmission range rn = √βln n/πn for some β < π/α, the LEARN routing protocol will not be able to find the route for any pair of nodes asymptotically almost sure. We also conducted simulations to study the performance of LEARN and compare it with a typical localized routing protocol (GPSR) and a global ad hoc routing protocol (DSR).