设lim(x→∞)[(1+x)/x]^(ax)=∫(-∞,a)te^tdx,求a

Question

设lim(x→∞)[(1+x)/x]^(ax)=∫(-∞,a)te^tdx,求a

Answer 1

lim(x→∞)[(1+x)/x]^(ax)
=lim(x→∞)([1+(1/x)]^(x) )^a
=e^a;
lim[1+(1/x)]^(x)=e是个重要的极限
∫(-∞,a)te^t dt
=∫(-∞,a)t d(e^t)
=t·e^t|(-∞,a) - ∫(-∞,a)e^t dt
=a·e^a - lim(t→-∞)t·e^t -e^t|(-∞,a)
=a·e^a - lim(t→-∞)t·e^t -e^a
令u=-t,则上式
=(a-1)·e^a+lim(u→+∞)u/e^u
=(a-1)·e^a+lim(u→+∞)1/e^u 【洛比达法则】
=(a-1)·e^a+0
=(a-1)·e^a
则(a-1)·e^a=e^a
a-1=1
a=2