Question

求 lim(x->∞)(ex/2+x^2*((1+1/x)^x - e))

Answer 1

原式=lim(x->∞) e^x/e^[ln(1+1/x)^x²]=lim(x->∞) e^x/e^x²ln(1+1/x)=lim(x->∞) e^(x-x²ln(1+1/x))=lim(x->∞) e^A
则lim(x->∞)A=lim(x->∞) [x-x²ln(1+1/x)]
令t=1/x，t-＞0，lim(x->∞)A=lim(x->∞)[(1/t-1/t²*ln(1+t)]=lim(x->∞) (t-ln(1+t))/t²=lim(x->∞) (1-1/(1+t))/2t=lim(x->∞) 1/[2(1+t)]=1/2
所以原式=e^(1/2)

Answer 2

lim（x→∞）[x-(x^2)ln(1+1/x)] (t=1/x,t→0)
=lim（t→0）[1/t-ln(1+t)/t^2]
=lim（t→0）[t-ln(1+t)]/t^2
=lim（t→0）[1-1/(1+t)]/(2t)
=lim（t→0）t/[(1+t)(2t)]
=1/2