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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, $\\Delta$. 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