lim（x→∞）[(3+x)/(6+x)]的[(x-1)/2]次方的极限

题目

lim（x→∞）[(3+x)/(6+x)]的[(x-1)/2]次方的极限

答案

为了简便,设1/t=-3/(x+6),则x=-3t-6
lim（x→∞）[(3+x)/(6+x)]^[(x-1)/2]
=lim[1-3/(x+6)]^[(x-1)/2]
=lim(1+1/t)^[(-3t-7)/2]
=lim1/[(1+1/t)^t)^(3/2)]*(1+1/t)^(-7/2)
=1/e^(3/2)