急！！！还是极限问题2

(4^n+5^n+6^n)^1/n ≤[3*6^n]^(1/n)

∴lim(4^n+5^n+6^n)^1/n ≤lim[3*6^n]^(1/n)=lim[3^(1/n)*6]=6

(4^n+5^n+6^n)^1/n ≥[6^n]^(1/n)

∴lim(4^n+5^n+6^n)^1/n ≥lim[6^n]^(1/n)=6

∴由夹逼定理可知lim(4^n+5^n+6^n)^1/n=6

技术不够

解法一：∵6<(6^n)^(1/n)<(4^n+5^n+6^n)^1/n<(6^n+6^n+6^n)^1/n=6*3^(1/n)
又因为lim(n->∞)[6*3^(1/n)]=6
∴由极限两边夹定理，知lim(n->∞)[(4^n+5^n+6^n)^1/n]=6
解法二：原式=lim(n->∞){6[(2/3)^n+(5/6)^n+1]^(1/n)}
=6lim(n->∞){[1+(2/3)^n+(5/6)^n]^[1/((2/3)^n+(5/6)^n)]}^[((2/3)^n+(5/6)^n)/n]
=6e^{lim(n->∞)[((2/3)^n+(5/6)^n)/n]}
=6e^0
=6
此题还可以用罗比达法则求解！