lim(x→0)(1-cosx)/(xarcsinx)=lim(1-cosx)/(xarcsinx)=1/2(x→0)

Question

lim(x→0) (1-cosx)/(xarcsinx)=_____
lim (1-cosx)/(xarcsinx)=1/2
(x→0)

Answer 1

lim (1-cosx)/(xarcsinx)
=lim[2sin^(x/2)]/(xarcsinx)
因为x→0时,sin(x/2)是无穷小量,等价于x/2
所以
=lim[2(x/2)(x/2)]/(xarcsinx)
=lim x/(2arcsinx)
=lim 1/[2/根号(1-x^2)] (分子分母求导)
=1/2