作者BaBi (迅雷不及掩耳盗铃)
看板trans_math
标题[极限] 夹挤定理求解
时间Wed Nov 20 02:35:52 2013
这篇不是要问问题啦XD
是我打算把数学板一些不错的题目慢慢地在这建档0.0
--
Show that lim xlnx = 0 without L'hospital Rule
x->0+
--
Let x = e^(-t)
lim xlnx = lim (-t)e^(-t) = - lim t/e^t
x->0+ t->∞ t->∞
Note that e^t > t^2/2, and t/e^t > 0 for x > 0
0 < t/e^t < 2/t, lim 2/t = 0
t->∞
By The Squeeze Theorem
lim xlnx = 0
x->0+
--
关於 e^t > t^2/2
令 f(x) = e^x - (x^2/2)
f'(x) = e^x - x > 0 for x > 0
故 f(x) 为一递增函数, 且 f(0) = e^0 - 0 = 1 > 0
f(x) = e^x - (x^2/2) > 0 在 x > 0 恒成立
故知 e^x > x^2/2
--
※ 发信站: 批踢踢实业坊(ptt.cc)
◆ From: 114.46.117.24
※ 编辑: BaBi 来自: 114.46.140.6 (11/20 11:42)