Question

高数，极限求解

Answer 1

对于原求极限部分我们可以转换为求x趋于+无穷大时[x^2/(x^2-1)]^x = e^[xln(x^2) - ln(x^2-1)]的极限。
当x趋于+无穷大时，x[ln(x^2) - ln(x^2-1)]的极限
＝[ln(x^2) - ln(x^2-1)]/(1/x) (0/0型）
=[2x/x^2 - 2x/(x^2-1)]/ (-1/x^2)
=(2x^3)[1/(x^2-1)-1/x^2)
= (2x^3)[x^2-(x^2-1)]/[x^2 (x^2-1)]
= 2x/(x^2-1) =0
因此原极限＝e^0=1

Answer 2

原式=lim(n->+∞) [1+1/(n^2-1)]^n
=lim(n->+∞) {[1+1/(n^2-1)]^(n^2-1)}^[n/(n^2-1)]
=e^0
=1