lim(x-->正无穷)[(x+1)ln(x+1)-(x+1)lnx]
=lim(x-->正无穷)(x+1)[ln(x+1)-lnx]
=lim(x-->正无穷)x[ln(x+1)-lnx]+lim(x-->正无穷)[ln(x+1)-lnx]
=lim(x-->正无穷)x[ln(x+1)/x]+lim(x-->正无穷)[ln(x+1)/x]
=lim(x-->正无穷)[ln(1+1/x)^x]+lim(x-->正无穷)[ln(1+1/x)]
=lne+ln1
=1