lim(x趋近于0) d((x-sinx)\xsinx) ÷ dx 怎么算?是求((x-sinx)\xsinx)的二阶导数后再求极限吗?

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lim(x趋近于0)d((x-sinx)\xsinx)÷dx怎么算?是求((x-sinx)\xsinx)的二阶导数后再求极限吗?lim(x趋近于0)d((x-sinx)\xsinx)÷dx怎么算?是求

lim(x趋近于0) d((x-sinx)\xsinx) ÷ dx 怎么算?是求((x-sinx)\xsinx)的二阶导数后再求极限吗?
lim(x趋近于0) d((x-sinx)\xsinx) ÷ dx 怎么算?是求((x-sinx)\xsinx)的二阶导数后再求极限吗?

lim(x趋近于0) d((x-sinx)\xsinx) ÷ dx 怎么算?是求((x-sinx)\xsinx)的二阶导数后再求极限吗?

d[(x-sinx)/(xsinx)]/dx
  = d(1/sinx - 1/x)/dx
  = -cosx/(sinx)^2 + 1/x^2
  = [(sinx)^2 -(x^2)cosx]/[(x^2)(sinx)^2],
故利用罗比达法则,有
  g.e.= lim(x→0)[(sinx)^2 -(x^2)cosx]/[(x^2)(sinx)^2]
 = lim(x→0)[(sinx)^2 -(x^2)cosx]/(x^4) (0/0)
 = lim(x→0)[2sinxcosx + (x^2)sinx - 2xcosx]/(4x^3)
   = (1/4)*lim(x→0)(sinx/x) + (1/2)lim(x→0)cosx*lim(x→0)[(sinx - x)/x^3]
  = 1/4 + (1/2)*lim(x→0)[(sinx - x)/x^3] (0/0)
  = 1/4 + (1/2)*lim(x→0)[(cosx - 1)/(3x^2)]
  = 1/4 + (1/2)*(1/6)
  = ……

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