limit問題 關於 e既law

lim(n→∞)_(1+1/n)^n
=?

Sol

lim(n→∞)_(1+1/n)^n

=lim(n→∞)_e^[nln(1+1/n)]

=e^(lim(n->∞)_[nln(1+1/n)])

=e^(lim(n->∞)_[ln(1+1/n)/(1/n)]

=e^(lim(x->0+)_[ln(1+x)/x] 0/0 type

=e^(lim(x->0+)_[1/(1+x)]

=e^1

=e



lim(n→∞)_(1─1/n)^n

=lim(n→∞)_e^[nln(1－1/n)]

=e^(lim(n->∞)_[nln(1－1/n)])

=e^(lim(n->∞)_[ln(1－1/n)/(1/n)]

=e^(lim(x->0+)_[ln(1－x)/x] 0/0 type

=e^(lim(x->0+)_[－1/(1－x)]

=e^(-1)

這裏可以幫到你
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lim (n->inf.) (1-1/n)^n

=lim (n->inf.) (1+1/(-n))^(-(-n))

=lim (n->inf.) 1/(1+1/(-n))^(-n)

=1/e^-1

=e//