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HMMT 二月 2005 · CALC 赛 · 第 5 题

HMMT February 2005 — CALC Round — Problem 5

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. Calculate ( ) x x x lim x − x .

x → 0

解析
  1. Calculate ( ) x x x lim x − x .

x → 0 Solution: − 1 x We first calculate lim + x : it is just exp(lim + x ln x ). But x → 0 x → 0 ln x 1 /x lim x ln x = lim = lim = lim − x = 0 2

x → 0 x → 0 x → 0 x → 0 1 /x − 1 /x x x x 1

by L’Hˆ opital’s Rule. Therefore lim x = 1. Then lim x = 0 = 0, so x → 0 x → 0 ( ) x x x lim + x − x = − 1. x → 0