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Semi-supervised and unsupervised machine learning methods often rely on graphs to model data, prompting research on how theoretical properties of operators on graphs are leveraged in learning problems. While most of the existing literature focuses on undirected graphs, directed graphs are very important in practice, giving models for physical, biological, or transportation networks, among many other applications. In this paper, we propose a new framework for rigorously studying continuum limits of learning algorithms on directed graphs. We use the new framework to study the PageRank algorithm, and show how it can be interpreted as a numerical scheme on a directed graph involving a type of normalized graph Laplacian. We show that the corresponding continuum limit problem, which is taken as the number of webpages grows to infinity, is a second-order, possibly degenerate, elliptic equation that contains reaction, diffusion, and advection terms. We prove that the numerical scheme is consistent and stable and compute explicit rates of convergence of the discrete solution to the solution of the continuum limit PDE. We give applications to proving stability and asymptotic regularity of the PageRank vector. Finally, we illustrate our results with numerical experiments and explore an application to data depth.