HMMT 二月 2004 · CALC 赛 · 第 3 题
HMMT February 2004 — CALC Round — Problem 3
题目详情
英文原题
- Find lim ( x + x − x − x ).
x →∞
解析
英文解析
- Find lim ( x + x − x − x ).
x →∞
Solution: 2 / 3
Observe that
[ ]
√
x/ 3 + 1 / 27
3 23
√ √
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 , whilethe denominator is at least 3 x , so the limit as x → ∞ is 0. By similar arguments,2
√
3 23
lim [( x − 1 / 3) − x − x ] = 0. So, the desired limit equalsx →∞
√ √
3 3
3 2 3 2
2 / 3 + lim [( x − 1 / 3) − x − x ] − lim [( x + 1 / 3) − x + x ] = 2 / 3 .
x →∞ x →∞