求lim(e^2+4^n+7^n)^(1/n)n趋近无穷大求该极限要详细解答

Question

求lim(e^2+4^n+7^n)^(1/n)
n趋近无穷大　　求该极限
要详细解答

Answer 1

求lim(e^2+4^n+7^n)^(1/n)可先求
lim 1/n*ln(e^2+4^n+7^n)
=lim 1/n*ln[(7^n)((e^2+4^n)/7^n+1)]
=lim [1/n*ln(7^n)+lim 1/n*ln(e^2/7^n+4^n/7^n+1)
=ln7+0=ln7
【对lim 1/n*ln(e^2/7^n+4^n/7^n+1)中
ln(e^2/7^n+4^n/7^n+1)的等价无穷小是e^2/7^n+4^n/7^n即4^n/7^n=(4/7）^n
∴lim 1/n*ln(e^2/7^n+4^n/7^n+1)=lim[（4/7）^n]/n=0】
∴lim(e^2+4^n+7^n)^(1/n)=e^ln7=7