Question

求极限lim乘以(x+1/x-1)*x （x趋向无穷） 求解答过程!急急急！谢谢！！！！！

Answer 1

解法一：原式=lim(x->∞)[(1+2/(x-1))^(((x-1)/2)(2x/(x-1)))]
={lim(x->∞)[(1+2/(x-1))^((x-1)/2)]}^{lim(x->∞)[2x/(x-1)]} (应用初等函数的连续性)
=e^{lim(x->∞)[2/(1-1/x)]} (应用重要极限lim(z->0)[(1+z)^(1/z)]=e)
=e^[2/(1-0)]
=e²；
解法二：原式=lim(x->∞){e^[x*ln((x+1)/(x-1))]} (应用对数性质)
=e^{lim(x->∞)[ln((x+1)/(x-1))/(1/x)]} (应用初等函数的连续性)
=e^{lim(x->∞)[2x²/(x²-1)]} (0/0型极限，应用罗比达法则并化简)
=e^{lim(x->∞)[2/(1-1/x²)]}
=e^[2/(1-0)]
=e²。

Answer 2

=lim [1+2/(x-1)]^x
=e^ lim x·ln[1+2/(x-1)]【取对数再取底】
=e^ lim 2x/(x-1)【等价无穷小代换】
=e^2

Answer 3