求极限lim(x→无穷）x(e^1/x-1)

Question

求极限lim(x→无穷）x(e^1/x-1)

Answer 1

令t=1/x,则t→0
lim x(e^(1/x)-1)
x→∞
=lim (1/t)·(e^t-1)
t→0
=lim(e^t-1)/t
t→0
根据等价无穷小：(e^x-1)=x
=lim t/t
t→0
=1