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

Answer 2

（1+x)\(1-x)=1+2x\(1-x)=1+2\(1\x-1)当x无穷时2\(1\x-1) 即1\X=0 2\(1\x-1)=-2 1-2=-1
(-1)^X X无穷时 可能为1或-1 所以极限不存在