Task admission is critical to delay-sensitive applications in mobile edge computing, but is technically challenging due to its combinatorial mixed nature and consequently limited scalability. We propose an asymptotically optimal task admission approach which is able to guarantee task delays and achieve (1-ϵ)-approximation of the computationally prohibitive maximum energy saving at a time-complexity linearly scaling with devices. ϵ is linear to the quantization interval of energy. The key idea is to transform the mixed integer programming of task admission to an integer programming (IP) problem with the optimal substructure by pre-admitting resource-restrained devices. Another important aspect is a new quantized dynamic programming algorithm which we develop to exploit the optimal substructure and solve the IP. The quantization interval of energy is optimized to achieve an [O(ϵ), O(1/ϵ)]-tradeoff between the optimality loss and time complexity of the algorithm. Simulations show that our approach is able to dramatically enhance the scalability of task admission at a marginal cost of extra energy, as compared with the optimal branch and bound method, and can be efficiently implemented for online programming.