limx趋近于1,x ^1/x-1等于?
问题描述：
limx趋近于1,x ^1/x-1等于?
答
lim_{x->1}x^[1/(x-1)]
=lim_{x->1}(1+x-1)^[1/(x-1)]
=lim_{x->1}[1+(x-1)]^[1/(x-1)]
=e最后答案不是e+1吗？e^(+1) = elim_{t->0}(1+t)^(1/t) = e,t = x-1.limx趋近于1时，lim1+(x-1) ^1/x-1=lim1+lim(x-1) ^1/x-1=1+e,难道不是这样的吗？？？？x->1, x-1->0啊, (x-1)^[1/(x-1)] 不趋于e. 只有当底趋于1,指数趋于无穷时,整个极限才可能趋于e. lim_{n->无穷}(1+1/n)^n = e.lim_{t->0}(1+t)^(1/t) = e.lim_{x->无穷}(1+1/x)^x = e.