求极限lim(n→∞)(2^n+3^n+5^n)^(1/n)=3Q

Question

求极限 lim (n→∞)(2^n+3^n+5^n)^(1/n) = 3Q

Answer 1

lim [n√(2^n+3^n+5^n)]e^{lim [(1/n)*ln(2^n+3^n+5^n)]}
对lim [(1/n)*ln(2^n+3^n+5^n)]用L'HOPITAL法则
lim [(1/n)*ln(2^n+3^n+5^n)]
=lim [(ln2*2^n+ln3*3^n+ln5*5^n)/(2^n+3^n+5^n)]
=lim {[ln2*(2/5)^n+ln3*(3/5)^n+ln5]/[(2/5)^n+(3/5)^n+1]}
当n→∞时(2/5)^n和(3/5)^n均趋向于0
故lim [1/n *ln(2^n+3^n+5^n)] = ln5
lim (n√2^n+3^n+5^n)
=e^{lim [(1/n)*ln(2^n+3^n+5^n)]}
=e^ln5
=5