Effective resistances are ubiquitous in graph algorithms and network analysis. In this work, we study sublinear time algorithms to approximate the effective resistance of an adjacent pair and . We consider the classical adjacency list model for local algorithms. While recent works have provided sublinear time algorithms for expander graphs, we prove several lower bounds for general graphs of vertices and edges: 1.It needs queries to obtain -approximations of the effective resistance of an adjacent pair and , even for graphs of degree at most 3 except and . 2.For graphs of degree at most and any parameter , it needs queries to obtain -approximations where is a universal constant. Moreover, we supplement the first lower bound by providing a sublinear time -approximation algorithm for graphs of degree 2 except the pair and . One of our technical ingredients is to bound the expansion of a graph in terms of the smallest non-trivial eigenvalue of its Laplacian matrix after removing edges. We discover a new lower bound on the eigenvalues of perturbed graphs (resp. perturbed matrices) by incorporating the effective resistance of the removed edge (resp. the leverage scores of the removed rows), which may be of independent interest.