Question

lim(x→0)（e的X次方-1）/x

Answer 1

x→0
lim
(e^x+x)^(1/x)
=lim
e^ln
(e^x+x)^(1/x)
=e^lim
ln
(e^x+x)^(1/x)
考虑
lim
ln
(e^x+x)^(1/x)
=lim
ln(e^x+x)
/
x
=lim
ln(1+e^x+x-1)
/
x
根据等价无穷小：ln(1+x)~x
=lim
(e^x+x-1)
/
x
=lim
(e^x-1)/x
+
lim
x/x
=1+lim
(e^x-1)/x
换元，t=e^x-1，x=ln(1+t)
=1+lim(t→0)
t
/
ln(1+t)
=1+lim
1
/
ln(1+t)^(1/t)
=1
+
1
/
ln
lim
(1+t)^(1/t)
根据重要的极限，
=1
+
1
/
lne
=1+1
=2
因此，原极限=e^2
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