求极限：用等价无穷小量.lim(x趋近于0负)(1-根号下cosx)tanx / (1-cosx)^3/2

题目

求极限：用等价无穷小量.lim(x趋近于0负)(1-根号下cosx)tanx / (1-cosx)^3/2

答案

lim(x→0) (1-√cosx)tanx / (1-cosx)^(3/2)
=lim (1-√cosx)(1+√cosx)tanx / (1-cosx)^(3/2)(1+√cosx)
=lim (1-cosx)tanx / (1-cosx)^(3/2)(1+√cosx)
=lim tanx / (1-cosx)^(1/2)(1+√cosx)
=lim tanx / √2*(1+√cosx)*sin(x/2)
根据等价无穷小：tanx~x,sin(x/2)~(x/2)
=lim 2x / √2*(1+√cosx)*x
=lim 2 / √2*(1+√cosx)
=1/√2
=√2/2
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