Question

求lim［（x+1）/（x+2）］^x，x趋向于无穷大的极限

Answer 1

lim [(x+1)/(x+2)]^x
x→∞
=lim {[1 -1/(x+2)]^[-(x+2)]}⁻¹·[1 -1/(x+2)]⁻²
x→∞
=e⁻¹·(1-0)⁻²
=e⁻¹·1
=1/e

Answer 2

(x+1)/(x+2) = 1 - 1/(x+2)
let
1/y = 1/(x+2)
x= y-2

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