截图来自同济高等数学第七版上册习题全解指南
(14)
lim(x->∞) ( 1+ a/x)^x = e^a
用洛必达法
y=1/x
lim(x->∞) ( 1+ a/x)^x
=lim(y->0) ( 1+ ay)^(1/y)
=lim(y->0) e^[ ln( 1+ ay)/y ] (0/0用洛必达法)
=lim(y->0) e^[ a/( 1+ ay) ]
=e^a
14. 不必一定用罗必塔法则吧。原式 = lim[(1+a/x)^(x/a)]^a = e^a
