Evaluate `lim_(xto0^(+)) x^(m)(logx)^(m),m,n,inN.`

Explanation

`underset(xto0^(+))limx^(m)(logx)^(n)" "(0xxoo" form")`
`=underset(xto0^(+))lim((logx)^(n))/(x^(-m))" "((oo)/(oo)" form")`
`=underset(xto0+)lim(n(logx)^((n-1))(1)/(x))/(-mx^(-m-1))" "`(Using L'Hospital's rule)
`=underset(xto0+)lim(n(logx)^(n-1))/(-mx^(-m))" "((oo)/(oo)" form")`
`underset(xto0+)lim(n(n-1)(logx)^((n-2))(1)/(x))/((-m)^(2)x^(-m-1))" "`(Using L'Hospital's rule)
`=underset(xto0+)lim(n(n-1)(logx)^(n-2))/(-m^(2)x^(-m))" "((oo)/(oo)" form")`
`=underset(xto0+)lim(n!)/((-m)^(n)x^(-m))=0" "("Differentiating "N^(r)" and "D^(r)" n times")`