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limsin2x sin3x

lim(sin2x/sin3x) =lim(xsin2x/xsin3x) =lim2/3(3xsin2x/2xsin3x) =lim2/3(sin2x/2x)/(sin3x/3x) =2/3lim(sin2x/2x)/lim(sin3x/3x) =2/3

直接用洛比达法则就可以了 原式=lim (2·cos2x)/(3·cos3x) =(2·cos2π)/(3·cos3π) =(2×1)/(3×(-1) ) = -2/3

当x→0时sinx→x所以limsin2x/sin3x=2x/3x=2/3

无穷小代换: 原式=lim (x-2x)/(x+3x) =lim -x/(4x) =-1/4

x→ 0时,sin3x~3x,所以原式=2/3

lim(x->0) sin2x/sin3x =lim(x->0) 2x/(3x) =2/3

解法一:等价无穷小 lim sin3x/sin2x x→0 =lim 3x/(2x) x→0 =3/2 解法二:高中解法,运用两个重要极限的第一个重要极限 lim sin3x/sin2x x→0 =lim (3/2)[sin3x/(3x)]/[sin2x/(2x)] x→0 =(3/2)·1/1 =3/2

如果题目是: lim(sin3x/x-xsin1/2x) x趋向于0 则 =lim (3cos3x )/1 - limx*sin(1/2x) =3-0 =3

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