We consider a one-dimensional infinite horizon calculus of variations problem (P), where the integrand is linear with respect to the velocity. The Euler-Lagrange equation, when defined, is not a differential equation as usual but reduces to an algebraic (or transcendental) equation C(x) = 0. Thus this first order optimality condition is not informative for optimal solutions with initial condition x(0) such that C(x(0)) not equal 0. To problem (P) we associate an auxiliary calculus of variations problem whose solutions connect as quickly as possible the initial conditions to some constant solutions. Then we deduce the optimality of these curves, called most rapid approach paths, for (P). According to the optimality criterium we consider, we have to assume a classical transversality condition. We observe that (P) possesses the turnpike property, the turnpike set being given by the preceding particular constant solutions of the auxiliary problem.