HMMT 二月 2004 · CALC 赛 · 第 3 题
HMMT February 2004 — CALC Round — Problem 3
题目详情
- Find lim ( x + x − x − x ). x →∞
解析
- Find lim ( x + x − x − x ). x →∞ Solution: 2 / 3 Observe that [ ] √ x/ 3 + 1 / 27 3 3 2 √ √ lim ( x + 1 / 3) − x + x = lim , 3 3 3 2 2 3 2 2 x →∞ x →∞ ( x + x ) + ( x + x )( x + 1 / 3) + ( x + 1 / 3) by factoring the numerator as a difference of cubes. The numerator is linear in x , while 2 the denominator is at least 3 x , so the limit as x → ∞ is 0. By similar arguments, √ 3 3 2 lim [( x − 1 / 3) − x − x ] = 0. So, the desired limit equals x →∞ √ √ 3 3 3 2 3 2 2 / 3 + lim [( x − 1 / 3) − x − x ] − lim [( x + 1 / 3) − x + x ] = 2 / 3 . x →∞ x →∞