lim(1+3^n)^1/n的值.n趋近于无穷.3^n就是3的n次方.(1+3^n)^1/n就是(1+3^n)的n分之1次方.

Question

lim(1+3^n)^1/n的值.n趋近于无穷.
3^n就是3的n次方.(1+3^n)^1/n就是(1+3^n)的n分之1次方.

Answer 1

lim[n→∞] (1+3^n)^(1/n)
=e^lim[n→∞] [ln(1+3^n)]/n
=e^lim[n→∞] 3^n*ln(3)/(1+3^n),洛必达法则
=e^ln(3)lim[n→∞] 3^n/(1+3^n)
=3lim[n→∞] 3^n/(1+3^n)
=3lim[n→∞] 3^n*ln(3)/[3^n*ln(3)]
=3lim[n→∞] 1
=3