Limit x1

40(a) R.H.S.

=1/ (1 + 1/(n-1) )

=1/[(n - 1 + 1)/(n-1)]

=(n - 1)/n

= 1 - 1/n

=L.H.S.

(b) lim(n->∞) (1 - 1/n)^(-n)

= lim(n->∞) {1/ [1 + 1/(n - 1) ]}^(-n) (by the result of (a))

= lim(n->∞) [1 + 1/(n - 1) }^n

= lim(n->∞) [1 + 1/(n - 1) ]^(n - 1) * [1 + 1/(n - 1) ]

= (e)(1)

= e

(c) lim(n->∞) (1 - 1/7n)^(n)

= lim(n->∞) {(1 - 1/7n)^(7n) }^(1/7)

= (1/e)^(1/7)





2011-02-19 16:28:24 補充：
lim(n->∞) [1 + 1/(n - 1) ]^(n - 1) = lim(n->∞) [1 + 1/n]^n = e