Question

lim x →∞ (x/x＋1)∧x 求大神解下，多谢

Answer 1

x →∞
lim (x/(x+1))^x
=lim (1 / (x+1)/x)^x
=lim 1 / (1+1/x)^x
根据极限的除法运算：
=1 / lim (1+1/x)^x
根据重要的极限：lim (1+1/x)^x=e
=1/e

有不懂欢迎追问

Answer 2

解： lim(x->∞)[(x/(x+1))^x]
=lim(x->∞){[(1+(-1)/(x+1))^((x+1)/(-1))]^((-x)/(x+1))}
={lim(x->∞)[(1+(-1)/(x+1))^((x+1)/(-1))]}^[lim(x->∞)((-x)/(x+1))]
=e^[lim(x->∞)((-x)/(x+1))] (应用重要极限lim(z->0)[(1+z)^(1/z)]=e)
=e^[lim(x->∞)((-1)/(1+1/x))]
=e^[(-1)/(1+0)]
=1/e。