Finding $\lim_{x\to 0} \frac {2\sin x-\sin 2x}{x-\sin x}$ geometrically

geometry calculus limits limits-without-lhopital

Not sure how convincing this is, but consider: enter image description here

That point in the middle of trangle $DBC$ is its centroid, so that all three subtriangles have the same area. Does the area of the upper right subtriangle approximate the area of the arc sliver to a high enough degree as the angle approaches $0$? If so, the ratio is of course $3$.

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