Question

高数。一个极限的问题

大家知道这个为什么不能这么做吗？求解

Answer 1

y->0

( 1+y)^(1/y) 

=e^[ln(1+y)/y]

= e^{ [ y -(1/2)y^2 +o(y^2)]/y  }

=e^[ 1 -(1/2)y +o(y)]

( 1+y)^(1/y) /e = e^[-(1/2)y +o(y)]

lim(x->+∞) ( 1+1/x)^(x^2)  /e^x

=lim(x->+∞) [( 1+1/x)^x /e  ]^x

y=1/x

=lim(y->0) [ ( 1+y)^(1/y) /e  ]^(1/y)

=lim(y->0) [ e^[-(1/2)y ]  ]^(1/y)

=e^(-1/2)