求极限limx-0y-0(sin(x^2y)-arcsinx^2y)/x^6y^3

Question

求极限limx-0y-0(sin(x^2y)-arcsinx^2y)/x^6y^3

Answer 1

想对y求极限：
y→0,[sin(x^2*y)-arcsin(x^2*y)]/(x^6*y^3)
→[x^2*cos(x^2*y)-x^2/√(1-x^4*y^2)]/(3x^6*y^2)
→[cos(x^2*y)-1]/(3x^4*y^2)
=-2[sin(x^2*y/2)]^2/(3x^4*y^2)
→-2(x^2*y/2)^2/(3x^4*y^2)
=-1/6.