Question

求大神帮忙解一道高数题

求大神帮忙解一道高数题越详细越好！拜谢

若以下回答无法解决问题，邀请你更新回答

Answer 1

x->0

分子
sinx~ x-(1/6)x^3
tanx ~ x+(1/3)x^3
sinx- tanx ~-(1/2)x^3
分母
(1+x^2)^(1/3) ~ 1+(1/3)x^2
(1+x^2)^(1/3) -1 ~ (1/3)x^2
sinx.[(1+x^2)^(1/3) -1 ] ~ (1/3)x^3
/
lim(x->0) ( sinx-tanx)/{ [(1+x^2)^(1/3) -1].[(1+sinx)^(1/2) -1] }
分子，分母同时乘以 [(1+sinx)^(1/2) +1]
=lim(x->0) ( sinx-tanx).[(1+sinx)^(1/2) +1]/{ [(1+x^2)^(1/3) -1].[(1+sinx) -1] }
=2lim(x->0) ( sinx-tanx)/{ [(1+x^2)^(1/3) -1]. (sinx) }
=2lim(x->0) -(1/2)x^3/[ (1/3)x^3]
=-3