Question

求【1+x)/(1-x)】的x次方，当x趋于无穷时的极限

Answer 1

(x->∞)lim
[(1+x)/(1-x)]^x
=(x->∞)lim
[(-1)(x+1)/(x-1)]^x
=(x->∞)lim
(-1)^x
[((x-1+2)/(x-1)]^x
=(x->∞)lim
(-1)^x
[1+2/(x-1)]^x
令(x-1)/2=t，则x=2t+1
=(x->∞)lim
(-1)^x
(t->∞)
lim
(1+1/t)^(2t+1)
=(x->∞)lim
(-1)^x
(t->∞)
lim
(1+1/t)^t(2+1/t)
利用重要极限(t->∞)lim(1+1/t)^t=e
=(x->∞)lim
(-1)^x
(t->∞)
lim
e^(2+1/t)
=(x->∞)lim
(-1)^x
e^2
当x为奇数时，原式=-e^2
当x为偶数时，原式=e^2
两种情况下极限不相等，所以原式不存在极限