Question

求当n趋向无限大时lim[(n+1)^2/(n-1)]的极限?

答案要详解!

Answer 1

你应该是问
lim[n→∞] (n+1)^[2/(n-1)]吧?
=lim[n→∞] (1+n)^[1/n·2n/(n-1)]
=1^lim[n→∞] 2n/(n-1)
=1^lim[n→∞] 2/(1-1/n),上下除n
=1^[2/(1-0)]
=1
这里运用了lim[x→∞] (1+x)^(1/x)=1
而不同于这lim[x→0] (1+x)^(1/x)=e

若是lim[n→∞] (n+1)²/(n-1)
=lim[n→∞] (n²+2n+1)/(n-1),上下除n²
=lim[n→∞] (1+2/n+1/n²)/(1/n-1/n²)
=(1+0+0)/(0-0)
=1/0
=∞,∴极限不存在