(cos(sin×)-cos×)/x^4极限

(cos(sin×)-cos×)/x^4极限

x->0
sinx ~ x - (1/6)x^3
cos(sinx)
~ cos( x - (1/6)x^3 )
~ 1 - (1/2)( x - (1/6)x^3 )^2 +(1/24)( x - (1/6)x^3 )^4
~ 1-(1/2)x^2 +(1/6)x^4 + (1/24)x^4
= 1-(1/2)x^2 +(5/24)x^4

cosx ~ 1 - (1/2)x^2 +(1/24)x^4
cos(sinx) -cosx
~ 1-(1/2)x^2 +(5/24)x^4 - [1 - (1/2)x^2 +(1/24)x^4]
=(1/6)x^4

lim(x->0) (cos(sinx) -cosx )/x^4
=lim(x->0) (1/6)x^4 /x^4
=1/6

x=cosθ(sinθ+cosθ)=(cosθ)^2+sinθcosθ y=sinθ(sinθ+cosθ)=(sinθ)^2+sinθcosθ 所以 x+y =1+2sinθcosθ (由倍角公式) =1+sin2θ 而 x-y =(cosθ)^2-(sinθ)^2 (由倍角公式) =cos2θ 因此有 (x+y-1)^2+(x-y)^2 =(sin2θ)^2+(cos2θ)^2 =1 即 (x+y-1)^2+(x-y)^2=1. 又因为 (x+y-1)^2+(x-y)^2=2(x^2+y^2)-2x-2y=1，即 x^2+y^2-x-y=0，所以上述参数方程代表的是一个圆：(x-1/2)^2+(y-1/2)^2=1/2. 圆心在(1/2,1/2)，半径为 根号2/2.