Question

求解一道高数极限题！

Answer 1

答案如下，望采纳，欢迎讨论

Answer 2

lim(n->∞) n[e(1+1/n)^(-n) -1]

let

y=1/x

consider

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)

e(1+y)^(-1/y) -1

=e{ e^[-1+(1/2)y]  + o(y) } -1

= e^[(1/2)y] -1 +o(y)

=(1/2)y +o(y)

lim(x->∞) x[e(1+1/x)^(-x) -1]

=lim(y->0) [e(1+y)^(-1/y) -1]/y

=lim(y->0) (1/2)y/y

=1/2

=> lim(n->∞) n[e(1+1/n)^(-n) -1] =1/2