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These methods allow us to extend previous nonvanishing results of Casarosa and Lambie-Hanson for $\\lim^k \\mathbf{A}$ for $k \\geq 2$. Specifically we show that, for a given $n$, it is relatively consistent with ZFC that $\\mathfrak{b} = \\mathfrak{d} = \\omega_n$ and $\\lim^k \\mathbf{A} \\ne 0$ whenever $1 \\leq k \\leq n$ (previously established with $2 \\leq k \\leq n$). 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