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For the classical age-dependent branching process $(h,G)$, where the offspring has probability generating function $h$ and all individuals have life-lengths independently picked from a distribution $G$, we focus on the setting $h = h_{\\alpha}^L$, with $L$ a function varying slowly at infinity and $\\alpha \\in (0,1)$. Here, $h^L_{\\alpha}(s) = 1 - (1-s)^{\\alpha} L(\\frac{1}{1-s}),$ as $s \\to 1$. 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For the classical age-dependent branching process $(h,G)$, where the offspring has probability generating function $h$ and all individuals have life-lengths independently picked from a distribution $G$, we focus on the setting $h = h_{\\alpha}^L$, with $L$ a function varying slowly at infinity and $\\alpha \\in (0,1)$. Here, $h^L_{\\alpha}(s) = 1 - (1-s)^{\\alpha} L(\\frac{1}{1-s}),$ as $s \\to 1$. 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