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HMMT 二月 2007 · CALC 赛 · 第 1 题

HMMT February 2007 — CALC Round — Problem 1

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 3 ] Compute:
    x 2
    x → 0 lim
    1 − cos( x )
解析

英文解析

  1. [ 3 ] Compute:
    x 2
    x → 0 lim
    1 − cos( x )
    Answer: 2 . Since sin ( x ) = 1 − cos ( x ), we multiply the numerator and denominator by 1 + cos( x )22
    and use the fact that x/ sin( x ) → 1, obtaining
    ( )
    2 22
    x x (1 + cos( x )) xlim = lim = lim · 2 = 2
    x → 0 x → 0 x → 02
    1 − cos( x ) 1 − cos ( x ) sin( x )
    x 2 x 2
    Remarks. Another solution, using L’Hˆ opital’s rule , is possible: lim = lim = 2.
    x → 0 x → 0
    1 − cos( x ) sin( x )