Question

用夹逼准则证明当x趋近于0时limcosx=1

帮忙证明下

Answer 1

0<=1-cosx=2sin²x/2<=2*(x/2)²=x²/2
因为
lim0=lim(x->0)x²/2=0
所以
由夹逼准则，得
当x趋近于0时运宴lim1-cosx=0

即
当x趋绝裤近于并悄简0时limcosx=1

Answer 2

lim(x->0)cosx
Taylor expansion of cosx
1-x^2/2!+x^4/6!-x^8/8!+...
for x->氏培0
cosx >= 1-x^2/孝宴2!
ie
1-x^2/2! <= cosx <巧核银=1
lim(x->0) cosx =1