A Riemannian metric on a compact manifold $M$ gives rise to a length function on the free loop space $LM$, whose critical points are the closed geodesics in the given metric on $M$. If $x$ is a homology class on $LM$, the "minimax" critical level $cr(x)$ is a critical value. Let $M$ be a sphere, and fix a metric and a coefficient field. We prove that the limit as $deg(x)$ goes to infinity of $cr(x)/deg(x)$ exists. Mark Goresky and Hans-Bert Rademacher are collaborators.