求(1/1-x)-(3/1-x三次方)的极限

来源：百度知道 编辑：UC知道 时间：2024/05/18 06:51:56

(1/1-x)-(3/1-x三次方)=(1/1-x)-[3/(1-x)(1+x+x^2)]=(1+x+x^2-3)/(1-x)(1-1-x-x^2)=(x-1)(x+2)/x(x-1)(x+1)=x+2/x^2+x=(1/x+2/x^2)/(1+1/x),当x趋于无穷大是，极限=0；x趋于1时 极限=1

x趋于1时，1/(1-x)-3/(1-x^3)=[(1+x+x^2)-3]/[1-x^2]=(x-1)(x+2)/[(1-x)(1+x+x^2)]=-(x+2)/(1+x+x^2)→-1(x→1).

lim(x->1) [1/(x-1) - 3/(1-x^3) ]
=lim(x->1) [1/(x-1) + 3/(x^3-1) ]
=lim(x->1) {1/(x-1) + 3/[(x-1)(x^2+x+1)] }
=lim(x->1) [(x^2+x+1) -3 ]/[(x-1)(x^2+x+1)]
=lim(x->1) (x^2+x-2)/[(x-1)(x^2+x+1)]
=lim(x->1) (x+2)(x-1)/[(x-1)(x^2+x+1)]
=lim(x->1) (x+2)/(x^2+x+1)
= (1+2)/(1+1+1)
=1