lim(h趋向于0)[sin(x+h)-sinx]/h

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lim(h趋向于0)[sin(x+h)-sinx]/hlim(h趋向于0)[sin(x+h)-sinx]/hlim(h趋向于0)[sin(x+h)-sinx]/hlim(h趋向于0)[sin(x+h)

lim(h趋向于0)[sin(x+h)-sinx]/h
lim(h趋向于0)[sin(x+h)-sinx]/h

lim(h趋向于0)[sin(x+h)-sinx]/h
lim(h趋向于0)[sin(x+h)-sinx]/h=(sinx)'=cosx

sin(x+h) - sinx = 2 cos(x+h/2) sin(h/2)
lim(h->0)[sin(x+h)-sinx]/h
= lim(h->0)[2 cos(x+h/2) sin(h/2)]/h
= lim(h->0 cos(x+h/2) * sin(h/2) / (h/2)
= cosx

sin(x+h)用x+h替换,sin(x)用x替换,可将h约去得1.