Question

LIM(X趋向0）（tanx-x）/（x-sinx）

Answer 1

LIM(X趋向0）（tanx-x）/x³=LIM(X趋向0）（sec²x-1）/(3x²)=LIM(X趋向0）（1-cos²x）/(3x²cos²x)

=LIM(X趋向0）（sinx/x)²/(3cos²x)=1/3
LIM(X趋向0）x³/(x-sinx)=LIM(X趋向0）3x²/(1-cosx)=LIM(X趋向0）3x²/2 sin²(x/2)
=LIM(X趋向0） 6 [(x/2)/sin(x/2)]²=6
原极限=（1/3）6=2

Answer 2

运用罗比达法则
原式=lim(x->0) (sec^2x-1)/(1-cosx)
=lim(x->0) (2sec^2xtanx)/(sinx)
=2*lim(x->0) 1/cos^3x
=2