Question

请教一个求极限的题

为啥这个极限等于3，咋算出来的捏🙏😗

Answer 1

lim(n->∞) [(2^n +3^n)/n ]^(1/n)
consider
L =lim(x->∞) [(2^x +3^x)/x ]^(1/x)
lnL
=lim(x->∞) ln[(2^x +3^x)/x ] /x (∞/∞ 分子分母分别求导)
=lim(x->∞){ [(ln2).2^x +ln3. 3^x ]/(2^x +3^x) -1/x }
=lim(x->∞) [(ln2).2^x +ln3. 3^x ]/(2^x +3^x)
分子分母同时除以3^x
=lim(x->∞) [(ln2).(2/3)^x +ln3 ]/[(2/3)^x +1]
=ln3
=> L =3
=>lim(n->∞) [(2^n +3^n)/n ]^(1/n) =3