Question

求极限 求(tanx-x)/(x^2*tanx)当x趋向0时候的极限值

求(tanx-x)/(x^2*tanx)当x趋向0时候的极限值

我很需要详细过程，谢谢啊

Answer 1

lim(tanx-x)/(x^2tanx)=
lim[(sinx/cosx)-x]/(x^2sinx/cosx)=
lim(sinx-x*cosx)/(x^2*sinx)【此步为上下同乘以cosx后所得】=
lim(sinx-x*cosx)/(x^3)【此步为利用x趋于0时，sinx和x等价无穷小】=
lim(cosx-cosx+x*sinx)/(3x^2)【此步为利用洛比达法则上下求导】=
lim(x*x)/(3x^2)【利用x趋于0时，sinx和x等价无穷小】=
1/3

Answer 2

lim(x->0)(tanx-x)/(x^2*tanx)

= lim(x->0)(tanx-x)/(x^2*x)*[x/tanx]

= lim(x->0)(tanx-x)/(x^3)

= lim(x->0)[(secx)^2-1]/(3x^2)

= 3lim(x->0)(tanx)^2/(x^2)

= 3

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lim(x->0)[tanx/x] = 1