230问答网 > x趋于0时，lim(x^2-sin^2 xcos^2 x)⼀(x^2sin^2 x)怎么转换成(x^2-(1⼀4)sin^2 2x)⼀x^4

x趋于0时，lim(x^2-sin^2 xcos^2 x)⼀(x^2sin^2 x)怎么转换成(x^2-(1⼀4)sin^2 2x)⼀x^4

2024-12-31 20:05:18

推荐回答（2个）

回答1：

2sinxcosx=sin2x
那么
sin^2xcos^2x=sin^22x/4
另外sinx等价于x，所以sin^2x等价于x^2,也即x^2sin^2x变成了x^4
不知您是否明白，望采纳！
若有不明还可问( ⊙ o ⊙ )啊！

回答2：

lim[x^2-(sin xcosx)^2]/(xsin x)^2
=lim[x^2-(sin2x)^2/4]/[x(1-cos2x)/2]
=lim[2x^2-(1-cos4x)/4]/[x(1-cos2x)]
=lim(8x^2-1+cos4x)/(4x-4xcos2x)
x→0,(8x^2-1+cos4x)→ 0 (4x-4xcosx)→ 0
=lim(x→0) (16x+4sin4x)/(4-4cos2x-8xsin2x)
x→0 (16x+4sin4x) → 0 ( 4-4cos2x-8xsin2x) → 0
=lim(x→0)(16+16cos4x)/(4+8sin2x-8sin2x-8xcos2x)
=32/4=8