Question

limx→∞[(2x+3)/(2x+1)]^(x+1)

Answer 1

[(2x+3)/(2x+1)]^(x+1)
=[(2x+2+1)/(2x+1)]^(x+1
=[1+2/(2x+1)]^(x+1)
2x+1=2t x=(2t-1)/2 x+1=(2t+1)/2
x→∞ t→∞
原式=根号(1+1/t)^(2t) * 根号(1+1/t)
=(1+1/t)^t*根号(1+1/t)
limx→∞[(2x+3)/(2x+1)]^(x+1)
=limt→∞(1+1/t)^t*[根号(1+1/t)]
=e*1=e

Answer 2

lim(x→∞)[(2x+3)/(2x+1)]^(x+1)
=lim(x→∞)[ 1+ 2/(2x+1)]^(x+1)
let
1/y = 2/(2x+1)
2x+1= 2y
2(x+1)=2y+1
x+1= (2y+1)/2
x->∞, y->∞
lim(x→∞)[ 1+ 2/(2x+1)]^(x+1)
=lim(y→∞)[ 1+ 1/y]^(y+1/2)
= lim(y→∞)[ 1+ 1/y]^(y) lim(y→∞)[ 1+ 1/y]^(1/2)
=e