求函数的极限lim((x→x/2)cosx)/(cos(x/2)-sin(x/2))
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求函数的极限lim((x→x/2)cosx)/(cos(x/2)-sin(x/2))
x->π/2吧
对分子
cosx=sin(π/2-x)
因为π/2-x ->0
所以
sin(π/2-x)~(π/2-x)
对分母
cos(x/2)-sin(x/2)
=√2[((√2)/2)cos(x/2)-((√2)/2)sin(x/2)]
=√2[sin(π/4)cos(x/2)-cos(π/4)sin(x/2)]
=√2 sin(π/4-x/2)
=√2 sin((π/2-x)/2)
√2(π/2-x)/2
所以原极限
=lim (π/2-x)/[√2(π/2-x)/2]
=√2
对分子
cosx=sin(π/2-x)
因为π/2-x ->0
所以
sin(π/2-x)~(π/2-x)
对分母
cos(x/2)-sin(x/2)
=√2[((√2)/2)cos(x/2)-((√2)/2)sin(x/2)]
=√2[sin(π/4)cos(x/2)-cos(π/4)sin(x/2)]
=√2 sin(π/4-x/2)
=√2 sin((π/2-x)/2)
√2(π/2-x)/2
所以原极限
=lim (π/2-x)/[√2(π/2-x)/2]
=√2
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