lim（x趋近于∞）（1+a/x）^bx+d=?

Question

lim（x趋近于∞）（1+a/x）^bx+d=?

Answer 1

答案应该是e^(ab)
lim(x->inf) (1 + a/x)^(bx+d)
= lim [1 + 1/(x*a^-1)]^[(x*a^-1) * 1/(x*a^-1) * (bx+d)]
= e^lim [1/(x*a^-1) * (bx+d)],重要极限lim(y->∞) (1+1/y)^y = e,这里的y = 1/(x*a^-1),前提是【1/(x*a^-1)】->0
= e^a*lim (bx+d)/x
= e^a*lim (b+d/x)
= e^a*(b+0)
= e^(ab)