求x→0,时(arctanx-sinx)/x^3极限arctanx等价无穷小是x,代进去得到(x-sinx)/x^3,用洛必达法则求出来极限是1/6和答案-1/6不一样为什么.

Question

求x→0,时(arctanx-sinx)/x^3极限
arctanx等价无穷小是x,代进去得到(x-sinx)/x^3,用洛必达法则求出来极限是1/6
和答案-1/6不一样为什么.

Answer 1

lim(x->0) ( arctanx - sinx )/ x^3 (0/0)
=lim(x->0) ( 1/(1+x^2) - cosx )/ (3x^2)
=lim(x->0) ( 1-(1+x^2)cosx )/ [(3x^2)(1+x^2)] (0/0)
=lim(x->0) ( -2xcosx + (1+x^2)sinx )/ [3(2x+4x^3)] (0/0)
=lim(x->0) ( -2(-xsinx+cosx) + 2x.sinx+ (1+x^2)cosx )/ [3(2+12x^2)] (0/0)
=(-2+1)/6
=-1/6