limX->0[1/ln(1+X)-1/X]极限......要用洛必达法则求

Question

limX->0 [ 1/ln(1+X) - 1/X ]
极限......要用洛必达法则求

Answer 1

原式=lim(x->0)[(x-ln(1+x))/(xln(1+x))]
=lim(x->0)[(1-1/(1+x))/(ln(1+x)+x/(1+x))] (0/0型极限,应用罗比达法则)
=lim(x->0)[x/((1+x)ln(1+x)+x)]
=lim(x->0)[1/(ln(1+x)+2)] (0/0型极限,应用罗比达法则)
=1/(0+2)
=1/2