Question

求极限：（2分之（n次根号下a+n次根号下b））的n次,当n趋于无穷时极限

Answer 1

原式=lim(n->∞){[(1+(a^(1/n)+b^(1/n)-2)/2)^(2/(a^(1/n)+b^(1/n)-2))]^[(a^(1/n)+b^(1/n)-2)/(2/n)]}
=e^{lim(n->∞)[(a^(1/n)+b^(1/n)-2)/(2/n)]} (应用重要极限lim(z->0)[(1+z)^(1/z)]=e)
=e^{lim(x->0)[(a^x+b^x-2)/(2x)]} (令x=1/n)
=e^{lim(x->0)[(a^xlna+b^xlnb)/2]} (0/0型极限,应用罗比达法则)
=e^[(lna+lnb)/2]
=√(ab).