The extremal functions of classes of matroids of bounded branch-width

For a set of matroids $\mathcal{M}$, let $ex_\mathcal{M}(n)$ be the maximum size of a simple rank-$n$ matroid in $\mathcal{M}$. We prove that, for any finite field $\mathbb{F}$, if $\mathcal{M}$ is a minor-closed class of $\mathbb{F}$-representable matroids of bounded branch-width, then $\lim_{n \rightarrow \infty} ex_\mathcal{M}(n) / n$ exists and is a rational number, $\Delta$. We also show that $ex_\mathcal{M}(n) - \Delta n$ is periodic when $n$ is sufficiently large and that $ex_\mathcal{M}$ is achieved by a subclass of $\mathcal{M}$ of bounded path-width.