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HMMT 二月 2007 · CALC 赛 · 第 1 题

HMMT February 2007 — CALC Round — Problem 1

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 3 ] Compute: 2 x lim x → 0 1 − cos( x )
解析
  1. [ 3 ] Compute: 2 x lim x → 0 1 − cos( x ) 2 2 Answer: 2 . Since sin ( x ) = 1 − cos ( x ), we multiply the numerator and denominator by 1 + cos( x ) and use the fact that x/ sin( x ) → 1, obtaining ( ) 2 2 2 x x (1 + cos( x )) x lim = lim = lim · 2 = 2 2 x → 0 x → 0 x → 0 1 − cos( x ) 1 − cos ( x ) sin( x ) 2 x 2 x Remarks. Another solution, using L’Hˆ opital’s rule , is possible: lim = lim = 2. x → 0 x → 0 1 − cos( x ) sin( x )