Question

1.limx→∞(1-1/2x)^x 2.limx→∞(1﹢x／x)^2x 3.limx→∞(1+1/x+3)^x 4.

Answer 1

1、

limx→∞ (1- 1/2x)^x

=limx→∞ [(1- 1/2x)^(-2x) ]^(-1/2)

显然limx→∞ (1- 1/2x)^(-2x)=e,

故limx→∞ [(1- 1/2x)^(-2x) ]^(-1/2) =e^(-1/2)

2、

limx→∞(1+x/x)^2x

=limx→∞ [(1+ 1/x)^x]^2 显然limx→∞ (1+ 1/x)^x=e

故原极限=e^2

3、

limx→∞(1+1/x+3)^x

=limx→∞(1+ 1/x+3)^[(x+3) *x/(x+3)]

=limx→∞ [(1+ 1/x+3)^(x+3)] ^ x/(x+3)

显然limx→∞ (1+ 1/x+3)^(x+3)=e,

而limx→∞ x/(x+3)=1

故原极限= e

4、

令1/x=t,

则

limx→0 (1+2x)^1/x

=limt→∞ (1+2/t)^ t

=limt→∞ [(1+2/t)^ t/2]^2

显然limt→∞ (1+2/t)^ t/2=e,

故原极限= e^2