lim(x→xπ)(tanx-sinx)/sin^3x题目错了 x→2π

lim(x→xπ)(tanx-sinx)/sin^3x题目错了 x→2π
lim(x→xπ)(tanx-sinx)/sin^3x
题目错了 x→2π

lim(x→xπ)(tanx-sinx)/sin^3x题目错了 x→2π
lim(tanx-sinx)/sin^3x
=limsinx(1/cosx-1)/sin^3x
=lim(1-cosx)/[sin^2x*cosx]
=lim2(sinx/2)^2/[4(sinx/2)^2(cosx/2)^2*cosx]
=lim1/[2(cosx/2)^2*cosx]
当x→2π时
cosx/2→ -1
cosx→ 1
原极限=1/2

化简(tanx-sinx)/sin^3x
=(1/cosx-1)/sin^2x
=1/cosx*[2sin^2(x/2)]/[4cos^2(x/2)*sin^2(x/2)]
=1/[2cosx*cos^2(x/2)]
得1/2