lim[cos(a/x)]^(x^2) x趋近于无穷大时求极限值

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lim[cos(a/x)]^(x^2) x趋近于无穷大时求极限值
lim[cos(a/x)]^(x^2) x趋近于无穷大时求极限值

lim[cos(a/x)]^(x^2) x趋近于无穷大时求极限值
lim[cos(a/x)]^(x^2)
=lim e^[x^2*lncos(a/x)]
=e^{lim x^2[cos(a/x)-1]}
=e^{lim x^2[-(a/x)^2/2]}
=e^[lim -a^2/2]
=e^(-a^2/2)

f(x)=[cos(a/x)]^(x^2)
lnf(X)=x^2ln[cos(a/x)]
令t=1/x
g(t)=lnf(x)=ln[cos(at)]/t^2 t趋向于0时
g(t)是0/0型
用罗必塔法则有
lim g(t)=-sin(at)*a/(2t*cos(at))=-a^2cosat/2=-a^2/2 ,t趋向于0
所以limf(x)= e^limg(t)=e^(-a^2/2)

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