Question

函数求极限lim[((1+x)^(1/x))/e]^(-1/x) x→0 求完整的步骤,

Answer 1

建议用Taylor展式：注意到u^v=e^(vlnu),以及ln(1+x)=x-x^2/2+o(x^2),因此
lim [(1+x)^(1/x)/e]^(-1/x)
=lim [e^(ln(1+x)/x)/e]^(-1/x)
=lim [e^(x-x^2/2+o(x^2)/x)/e]^(-1/x)
=lim [e^(-x/2+o(x))]^(-1/x)
=lim e^(1/2+o(1))
=e^(1/2).
此题可以取对数用用洛必达法则,但比较繁琐.