lim
(x-arcsinx)/(sinx)^3 (分母等价无穷小代换）
= lim
(x-arcsinx)/x^3 (0/0,罗必塔）
= lim
[1-1/√(1-x^2)]/(3x^2)
= lim
[√(1-x^2)-1]/[(3x^2)√(1-x^2)]
= lim
[√(1-x^2)-1]/[(3x^2)]
= lim
[(1-x^2)-1]/{[(3x^2)][√(1-x^2)+1]}
= lim
(-1)/{3[√(1-x^2)+1]} = -1/6.
2.lim
(tanx-sinx)/(xsinx^2)
= lim
tanx(1-cosx)/(xsinx^2) (等价无穷小代换）
= lim
(x*x^2/2)/x^3 = 1/2.
3.z=e^(x^2+y^2),z'
=2xe^(x^2+y^2),z'
=2ye^(x^2+y^2),
在点(1,0),z'
=2e,z'
=0,dz=2edx.