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/xx趋近于+0求极限lim(cos√x)^π/xx趋近于+0求极限lim(cos√x)^π/x令√x=t,则x→0时,t→0原式=lim(cost)^(π/

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

x趋近于+0求极限lim(cos√x)^π/x
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)