Question

lim（ n趋近于无穷）[(n次根号下a+n次根号下b+n次根号下c）/3]n次方,a>0,b>0,c>0

Answer 1

∵lim(n->∞){n*ln[(a^(1/n)+b^(1/n)+c^(1/n))/3]}

=lim(n->∞){ln[(a^(1/n)+b^(1/n)+c^(1/n))/3]/(1/n)}

=lim(x->0){ln[(a^x+b^x+c^x)/3]/x} (令x=1/n)

=lim(x->0){[ln(a^x+b^x+c^x)-ln3]/x}

=lim(x->0)[(a^x*lna+b^x*lnb+c^x*lnc)/(a^x+b^x+c^x)] (0/0型极限,应用罗比达法则)

=(lna+lnb+lnc)/(1+1+1)

=ln(abc)/3

∴lim(n->∞){[(a^(1/n)+b^(1/n)+c^(1/n))/3]^n}

=lim(n->∞){e^{n*ln[(a^(1/n)+b^(1/n)+c^(1/n))/3]}}

=e^{lim(n->∞){n*ln[(a^(1/n)+b^(1/n)+c^(1/n))/3]}}

=e^[ln(abc)/3]

=(abc)^(1/3).