Given an $x_0\in R^n$ we study the infinite horizon problem of minimizing the expression $\int_0^Tf(t,x(t),x'(t))dt$ as $T$ grows to infinity where $x:[0,\infty)\to R^n$ satisfies the initial condition $x(0) = x_0$ . We analyse the existence and the properties of approximate solutions for every prescribed initial value $x_0$ . We also establish that for every bounded set $E\subset R^n$ the $C([0,T])$ norms of approximate solutions $x:[0,T]\rightarrow R^n$ for the minimization problem on an interval $[0,T]$ with $x(0),x(T)\in E$ are bounded by some constant which does not depend on $T$ .