计算极限lim（x→∞）[x-(x^2)ln(1+1/x)]

题目

计算极限lim（x→∞）[x-(x^2)ln(1+1/x)]

答案

lim（x→∞）[x-(x^2)ln(1+1/x)] (t=1/x,t→0)
=lim（t→0）[1/t-ln(1+t)/t^2]
=lim（t→0）[t-ln(1+t)]/t^2
=lim（t→0）[1-1/(1+t)]/(2t)
=lim（t→0）t/[(1+t)(2t)]
=1/2