Question

求详细讲解几道微积分题目，希望说下解题的方法

limx->无穷（x+1/x+2)x次方 limx->1(1-2lnx)1/lnx次方

Answer 1

详细解答与说明，请参见图片。

点击放大，再点击再放大：

Answer 2

①lim(x->∞) [(x+1)/(x+2)]^x

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

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

②lim(x->1) (1-2lnx)^(1/lnx)

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

=lim(x->1) { [1+(-2lnx)]^[1/(-2lnx)] } ^ (-2)
∵lim(x->1) -2lnx = 0 , ∴ lim(x->1) { [1+(-2lnx)]^[1/(-2lnx)] } = e
= e^(-2)