Question

lim(x→∞)e^x/[(1+1/x)^x^2]求极限,

Answer 1

=lim(x→∞)e^x / e^x^2·ln[(1+1/x)]

=e^ lim(x→∞) (x - x^2·ln[(1+1/x)])

令u=1/x,则u→0.

原式=e^ lim(u→0) (1/u - ln[(1+u)] /u²)

=e^ lim(u→0) ( (u - ln[(1+u)] ) /u²)

=e^ lim(u→0) ( (1 - 1/(1+u) ) /2u)

=e^ lim(u→0) ( 1/[2(1+u)] )

=e^(1/2)

即√e