We consider the fixed points of nonlinear operators that naturally arise in games and general equilibrium models with endogenous networks, dynamic programming, in models of opinion dynamics with stubborn agents, and financial networks. We study limit cases that correspond to high coordination motives, infinite patience, vanishing stubbornness, and small exposure to the real sector in the applications above. Under monotonicity and continuity assumptions, we provide explicit expressions for the limit fixed points. We show that, under differentiability, the limit fixed point is linear in the initial conditions and characterized by the Jacobian of the operator at any constant vector with an explicit and linear rate of convergence. Without differentiability, but under additional concavity properties, the multiplicity of Jacobians is resolved by a representation of the limit fixed point as a maxmin functional evaluated at the initial conditions. In our applications, we use these results to characterize the limit equilibrium actions, prices, and endogenous networks, show the existence and give the formula of the asymptotic value in a class of zero-sum stochastic games with a continuum of actions, compute a nonlinear version of the eigenvector centrality of agents in networks, and characterize the equilibrium loss evaluations in financial networks.