1.limx→∞(1-1/2x)^x2.limx→∞(1﹢x／x)^2x3.limx→∞(1+1/x+3)^x4.limx→0(1+2x)^1/x

Question

1.limx→∞(1-1/2x)^x 2.limx→∞(1﹢x／x)^2x 3.limx→∞(1+1/x+3)^x 4.limx→0(1+2x)^1/x

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