求极限的题目,求过程

题目

求极限的题目,求过程
x趋于无穷,lim[x-x^2In(1+1/x)]等于多少

答案

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