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Consider $f \in L^{2}(0,1)$ and $g \in L^{\infty}(0,1)$ such that $ \text{lim} ~g(x) = 0 \ \ \text{when} \ \ x \to 0^{+};$ $g(x) > 0 \ \forall x \in (0,1)$; $\text{lim}~\dfrac{g(x)}{x^{\alpha}} =...