HMMT 二月 2005 · CALC 赛 · 第 5 题

英文解析

5. Calculate
   ( )
   x xxlim x − x .
   x → 0+
   Solution: − 1
   We first calculate lim + x : it is just exp(lim + x ln x ). Butxx → 0 x → 0
   ln x 1 /xlim x ln x = lim = lim = lim − x = 0 + + + +2
   x → 0 x → 0 x → 0 x → 0
   1 /x − 1 /xxx x 1 + +
   by L’Hˆ opital’s Rule. Therefore lim x = 1. Then lim x = 0 = 0, sox → 0 x → 0
   ( )
   x xxlim + x − x = − 1.
   x → 0