题目

当x—>a时,lim (e^x-e^a)/(x-a) 求极限
没学洛必达法则 和导数

提问时间：2020-11-30

答案

Ans 1：洛必达法则
lim(x→a) (e^x - e^a)/(x - a)
= lim(x→a) (e^x - 0)/(1 - 0)
= lim(x→a) e^x
= e^a
Ans 2：导数定义
lim(x→a) (e^x - e^a)/(x - a)
= [e^x]' |(x = a)
= e^x |x = a
= e^a
Ans 3：用lim(x→0) (e^x - 1)/x = 1
lim(x→a) (e^x - e^a)/(x - a)
= lim(x→a) e^a(e^x/e^a - 1)/(x - a)
= lim(x→a) e^a[e^(x - a) - 1]/(x - a),令u = x - a,x→a时u→0
= lim(u→0) e^a(e^u - 1)/u
= e^a • 1
= e^a