limx趋于0(tanx-sin)/x^3用洛比达法则

limx趋于0(tanx-sin)/x^3用洛比达法则
limx趋于0(tanx-sin)/x^3用洛比达法则

limx趋于0(tanx-sin)/x^3用洛比达法则
lim (tanx-sinx)/x³
=lim (1/cos²x-cosx)/3x²
=lim (2sinx/cos³x+sinx)/6x
=lim ((2+4sin²x)/(cosx)^4+cosx)/6
=lim (2+1)/6
=1/2

原式=lim(x-->0)[(sinx/cosx-sinx)/sin(x^3)]
=lim(x-->0)(sinx/cosx)(1-cosx)/sin(x^3)]
=lim(x-->0)[(x/cosx)(x^2/2)/x^3]
=lim(x-->0)[(x^3/2cosx)/x^3]
=lim(x-->0)[1/2cosx]=1/2