高等数学慎入

令Y=√x，利用（tgY）/Y=（sinY）/Y=1,(1-cosY)/Y^2=1/2;
l i m 1-√cosx
------------
x→0+ x(1-cos√x)

l i m 1-√cosY^2
= ------------
Y→0+ Y^2*(1-cosY)

l i m Y*tgY^2
= ------------
Y→0+ 2Y*(1-cosY) +Y^2*sinY

l i m (tgY^2)/Y^2
= ------------
Y→0+ 2(1-cosY)/Y^2 +(sinY)/Y

=1/2

1-cos√x=2(sin(√x/2))^2半角公式
当x→0+时,sin(√x/2)/(√x/2)极限为1
故√x/2是sin(√x/2)的同阶无穷小量即√x/2 ~ sin(√x/2)
所以x→0+时,1-cos√x=2*(sin(√x/2))^2=2*(√x/2)^2=x/2
代入分母得1/2*x^2

分子分母同时乘以1+√cosx
分子变成1-cosx ~ 1/2*x^2

结果是1/(1+√cosx )=1/2

1-sqrt(cos(x)) = 1 - sqrt(1-1/2x^2) = 1-(1-(1/2)*1/2x^2)=1/4x^2