Question

(急）极限问题:x趋近于0正,求[(1+x)^(1/x)/e]^(1/x)的极限

Answer 1

(1+x)^(1/x)=e只是极限状态下成立,

如果可以随便代的话lim(1+x)^(1/x) = (1+0)^(1/x)=1,显然错误.

x趋近于0正,lim[(1+x)^(1/x)/e]^(1/x)

=

x趋近于正无穷,lim[(1+1/x)^(x)/e]^x

=

lim[(1+1/x)^(xx) / e^x

=

lim e ^ (xxln(1+1/x) - x)

=

e ^ (lim(xxln(1+1/x) - x))

指数的极限用洛必达法则