lim(cos1/x)^x^2limX趋于无穷(cos1/x)^x^2 x趋于无穷,结果是答案是e^-0.5

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lim(cos1/x)^x^2limX趋于无穷(cos1/x)^x^2 x趋于无穷,结果是答案是e^-0.5
lim(cos1/x)^x^2
limX趋于无穷(cos1/x)^x^2 x趋于无穷,结果是答案是e^-0.5

lim(cos1/x)^x^2limX趋于无穷(cos1/x)^x^2 x趋于无穷,结果是答案是e^-0.5
这是个1^∞极限,先取自然对数
lim(x→∞)ln(cos1/x)^x^2
=lim(x→∞)x^2ln(cos1/x) (1/x=t,t→0)
=lim(t→0)ln(cost) /t^2 (运用洛必达法则)
=lim(t→0)(-sint/cost) /(2t)
=lim(t→0)-sint /(2t)=-1/2
所以
lim(x→∞)(cos1/x)^x^2=lim(x→∞)e^ln(cos1/x)^x^2=e^(-1/2)

lim(t→0)e^ln(cost) /t^2 ,用罗比达法则得到lim(t→0)e^(-sint /(2t))=e^-1/2

答案是e^-0.5