x趋近于+0求极限lim(cos√x)^π/x
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x趋近于+0求极限lim(cos√x)^π/x
![x趋近于+0求极限lim(cos√x)^π/x](/uploads/image/z/33309-45-9.jpg?t=x%E8%B6%8B%E8%BF%91%E4%BA%8E%2B0%E6%B1%82%E6%9E%81%E9%99%90lim%EF%BC%88cos%E2%88%9Ax%EF%BC%89%5E%CF%80%2Fx)
令√x = t,则x→0时,t→0
原式= lim (cost)^(π/t²)
=lim (1 + (cost-1) )^(π/t²)
=lim (1 + (cost-1) )^(1/(cost-1) * (cost-1)π/t²)
=lim ( (1 + (cost-1) )^(1/(cost-1)) )^ (cost-1)π/t²)
注意到 cost-1与 -t²/2等价
所以原式 = e^(-π/2)
原式= lim (cost)^(π/t²)
=lim (1 + (cost-1) )^(π/t²)
=lim (1 + (cost-1) )^(1/(cost-1) * (cost-1)π/t²)
=lim ( (1 + (cost-1) )^(1/(cost-1)) )^ (cost-1)π/t²)
注意到 cost-1与 -t²/2等价
所以原式 = e^(-π/2)
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