In this paper first, we propose a formula to lift a connection on $M$ to its higher order tangent bundles $T^rM$, $r\in\N$. More precisely, for a given connection $\nabla$ on $T^rM$, $r\in\N\cup\{0\}$, we construct the connection $\nabla^c$ on $T^{r+1}M$. Setting $\nabla^{c_i}={\nabla^{c_{i-1}}}^c$, we show that $\nabla^{c_\infty}=\vlim \nabla^{c_i}$ exists and it is a connection on the Fr\'{e}chet manifold $T^\infty M=\vlim T^iM$ and the geodesics with respect to $\nabla^{c_\infty}$ exist. In the next step, we will consider a Riemannian manifold $(M,g)$ with its Levi-Civita connection $\nabla$. Under suitable conditions this procedure gives a sequence of Riemannian manifolds $\{(T^iM$, $g_i)\}_{i\in\N}$ equipped with a sequence of Riemannian connections $\{\nabla^{c_i}\}_{i\in\N}$. Then we show that $\nabla^{c_\infty}$ produces the curves which are the (local) length minimizer of $T^\infty M$.
