[(x+2)ln(x+2)-2(x+1)ln(x+1)+xln(x)]*x x趋向于正无穷,求它的极限.怎么解,答案是1

来源：学生作业帮编辑：神马作文网作业帮分类：数学作业时间：2024/11/18 17:24:24

(x->+∞)lim[(x+2)ln(x+2)-2(x+1)ln(x+1)+xln(x)]*x
=(x->+∞)[(x+2)ln(x+2)-2(x+1)ln(x+1)+xln(x)]/(1/x),【为0/0型】用罗比达法则
=(x->+∞)-[ln(x+2)+1-2ln(x+1)-2+ln(x)+1]/(1/x^2),
=(x->+∞)lim-[ln(x+2)-2ln(x+1)+ln(x)]/(1/x^2),(0/0型)
=(x->+∞)lim(1/2)[1/(x+2)-2/(x+1)+1/x]/(1/x^3),
=(x->+∞)lim(1/2)[1/(x+2)-2/(x+1)+1/x]/(1/x^3)
=(x->+∞)lim(1/2)[2/[x(x+1)(x+2)]/(1/x^3)
=(x->+∞)lim[x^3/[x(x+1)(x+2)]
=1

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