limx→0 (x log x) = 0 and  limx→0 1/log x = 0. Let m, n ∈ N. i. Using L’Hˆopital’s rule, prove limx→0 [xm (log x)n ] = − (m/n) limx→0 [xm (log x)n+1].  Hence, by induction or otherwise, prove limx→0 [xm (log x)n ] = 0. ii. Manipulate  ∫01  xm(log x)n dx into an integral of similar form but with a reduced power or powers in the integrand. iii. Then compute  ∫01 x6 (log x)5 dx.

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