lim(x趋于0)(tanx-sinx)/3x^2

0/0型,用洛比达法则,
lim(x趋于0)(tanx-sinx)/(3x^2)=lim(x趋于0)(tanx-sinx)'/(3x^2)'=lim(x趋于0)[(tanx)'-(sinx)']/(6x)
=lim(x趋于0)[1/(cosx)^2-cosx]/(6x)==lim(x趋于0)[1/(cosx)^2-cosx]'/(6x)'
=lim(x趋于0){[1/(cosx)^2]'-(cosx)'}/6=lim(x趋于0)[(2sinx)/(cosx)^3+sinx]/6=[(2sin0)/(cos0)^3+sin0]/6=0

那我就不用洛必达法则了,用定理lim[x→0] sinx/x=1
lim[x→0] (tanx-sinx)/3x²
= (1/3)lim[x→0] (sinx/cosx-sinx)/x²
= (1/3)lim[x→0] (sinx-sinxcosx)/(x²cosx)
= (1/3)lim[x→0] sinx(1-cosx)/(x
...

(tanx-sinx)/3x^2~x(1/cosx-1)/3x^2
~(1-cosx)/3xcosx~2[sin(x/2)]^2/3xcosx=0