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求：当n趋于无穷时,lim(cosa/2)*(cosa/4)*……(cosa/2^n)求思路
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求：当n趋于无穷时,lim (cosa/2)*(cosa/4)*……(cosa/2^n) 求思路
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答案和解析
=lim (cosa/2)*(cosa/4)*……( cosa/2^(n-1) )*(cosa/2^n)·(sina/2^n) / (sina/2^n)
=lim (cosa/2)*(cosa/4)*……( cosa/2^(n-1) )·( (1/2)sina/2^(n-1) ) / (sina/2^n)
=……
=lim (1/2^n)sina / (sina/2^n)
即 lim sina / [(2^n)·(sina/2^n)]
由等价无穷小,则n趋于无穷时,a/2^n→0.则 sina/2^n a/2^n
则lim sina / [(2^n)·(sina/2^n)]
=lim sina / [(2^n)·(a/2^n)]
= sina / a
=lim (cosa/2)*(cosa/4)*……( cosa/2^(n-1) )·( (1/2)sina/2^(n-1) ) / (sina/2^n)
=……
=lim (1/2^n)sina / (sina/2^n)
即 lim sina / [(2^n)·(sina/2^n)]
由等价无穷小,则n趋于无穷时,a/2^n→0.则 sina/2^n a/2^n
则lim sina / [(2^n)·(sina/2^n)]
=lim sina / [(2^n)·(a/2^n)]
= sina / a
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