n趋近于无穷大,求(1+2^n)^(1/n)的极限

题目

n趋近于无穷大,求(1+2^n)^(1/n)的极限

答案

lim(n→∞) (1+2^n)^(1/n)
=lim(n→∞) e^{ln[(1+2^n)^(1/n)]}
=lim(n→∞) e^[ln(1+2^n)/n]
=e^ lim(n→∞) [ln(1+2^n)/n]
=e^ lim(n→∞) [(2^n)*ln2/(1+2^n)]
=e^(ln2)=2