Question #8e58a
1 Answer
Explanation:
Using the limit definition of
One way to define
Using this definition, we have
#=(lim_(n->oo)(1+1/n)^n)^2#
(The above step follows from
#=e^2#
Using logarithms and L'Hopital's rule
#=lim_(n->oo)e^(2nln(1+1/n))#
#=e^(lim_(n->oo)2nln(1+1/n))#
(The above step follows from
#=lim_(n->oo)(d/(dn)2ln(1+1/n))/(d/(dn)1/n)#
*(The above step follows from L'Hopital's rule. While it does not technically apply to discrete sequences, we could consider the continuous analog and apply L'Hopital's rule to get the same result.)
#=lim_(n->oo)(2/(1+1/n)(-1/n^2))/(-1/n^2)#
#=lim_(n->oo)2/(1+1/n)#
#=2/(1+0)#
#=2#
#=e^2#
