lim(n趋向无穷){[(n^2+1)^(1/2)]/(n+1)}^n怎么解.
问题描述：
lim(n趋向无穷){[(n^2+1)^(1/2)]/(n+1)}^n怎么解.
答
lim(n->∞){[(n^2+1)^(1/2)]/(n+1)}^n=lim(n->∞)[(n^2+1)/(n^2+2n+1)]^(n/2)]=
lim(n->∞)[(1-2n/(n^2+2n+1)]^(n/2)]=
lim(n->∞)(1-2n/(n^2+2n+1)^{-(n^2+2n+1)/2n*[-n^2/(n^2+2n+1)]}=e^(-1)
答
lim e^{n[ln√(n²+1)-ln(n+1)]}
=lim e^{[ln√(n²+1)-ln(n+1)]/(1/n)} 应用洛必达法则
=lim e^[-(n-1)n²/(n+1)(n²+1)]
=1/e
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