Lim [x^2/(x^2-1)]^x (x→∞)

来源：学生作业帮助网编辑：作业帮时间：2024/05/01 18:42:06

Lim [x^2/(x^2-1)]^x (x→∞)
Lim [x^2/(x^2-1)]^x (x→∞)

Lim [x^2/(x^2-1)]^x (x→∞)
lim [x^2/(x^2-1)]^x (x→∞)
=lim1/[(1-x^(-2))^x]
=lim1/[(1-1/x)^x*(1+1/x)^x]
=lim1/[(1-1/x)^x]*lim1/[(1+1/x)^x]
=1/lim[(1-1/x)^x]*1/lim[(1+1/x)^x]
=1/e^(-1)*1/e^1
=1

取对数：原式=e^(x*lnx^2/x^2-1)
又因为：x*lnx^2/x^2-1=x(lnx^2-lnx^2-1)
然后用罗比达法则求导。

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