202122: l'Hôpital's rule, integrals. You are given that limx→o (x log x) = 0 and that limx→0 logx = 0. Let m, n € 4. a) Use L'Hôpital's rule to show that lim [xm (log x)"] X→0 m n x→0 lim [xm (log x)"+1]. Hence, by induction or otherwise, show that lim [x (log x)"] = 0. X→0 b) Manipulate xm (logx)" dx into an integral of similar form but with a reduced power or powers in the integrand. c) Hence compute ¹ x6(log x)5 dx.

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Q: properties of logarithms to

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