Question

当n趋向于∞时,求（2的N次方+3的N次方+5的N次方）开N次方的极限

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