求当x趋于0时lim(1+sinx)^(2/x);lim1-cos2x/(sin^2)3x的极限;当x趋于无穷时lim(

求当x趋于0时lim(1+sinx)^(2/x);lim1-cos2x/(sin^2)3x的极限;当x趋于无穷时lim(x-1)(sin1/x-1)的极限
yii291 1年前 已收到2个回答 举报

sydfz 幼苗

共回答了21个问题采纳率:85.7% 举报

lim(1+sinx)^(2/x)=e^2
lim1-cos2x/(sin^2)3x=1/18
lim(x-1)(sin1/x-1)=1

1年前 追问

6

yii291 举报

可以写下过程吗?谢谢啦!

举报 sydfz

lim(x→0) (1+sinx)^(2/x) =lim(x→0) [(1+sinx)^(1/sinx)]^(2sinx/x) =lim(x→0) [(1+sinx)^(1/sinx)]^[lim(x→0) (2sinx/x)] =e^2 lim(x→0) 1-cos2x/(sin^2)3x =lim(x→0) (x^2/2)/(3x)^2 =1/18 lim(x→∞) (x-1)(sin1/x-1) =lim(x→∞) (sin1/x-1)/[1/(x-1)] =1

rick98 幼苗

共回答了4个问题 举报

①考察两个重要极限
lim(1+sinx)^2/x
=lim(1+sinx)^(1/sinx*sinx*2/x)
=e^lim(2sinx/x)
=e²
②考察等价无穷小
x→0时,1-cosx~x²/2,sinx~x
∴lim(1-cos2x)/sin²3x
...

1年前

3
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