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For a graph class $\mathcal{F}$, let $ex_{\mathcal{F}}(n)$ denote the maximumnumber of edges in a graph in $\mathcal{F}$ on $n$ vertices. We show that forevery proper minor-closed graph class $\mathcal{F}$ the function$ex_{\mathcal{F}}(n) - \Delta n$ is eventually periodic, where $\Delta =\lim_{n \to \infty} ex_{\mathcal{F}}(n)/n$ is the limiting density of$\mathcal{F}$. This confirms a special case of a conjecture by Geelen, Gerardsand Whittle. In particular, the limiting density of every proper minor-closedgraph class is rational, which answers a question of Eppstein.  As a major step in the proof we show that every proper minor-closed graphclass contains a subclass of bounded pathwidth with the same limiting density,confirming a conjecture of the second author.  Finally, we investigate the set of limiting densities of classes of graphsclosed under taking topological minors.

Densities of minor-closed graph classes are rational