HMMT 二月 2008 · CALC 赛 · 第 4 题

英文解析

4. [ 4 ] Let a, b be constants such that lim = 1. Determine the pair ( a, b ).
   x → 12
   x + ax + b
   Answer: ( − 2 , 1) When x = 1, the numerator is 0, so the denominator must be zero as well, so
   1 + a + b = 0. Using l’Hˆ opital’s rule, we must have
   (ln(2 − x )) 2 ln(2 − x )2
   1 = lim = lim ,
   x → 1 x → 12
   x + ax + b ( x − 2)(2 x + a )
   and by the same argument we find that 2 + a = 0. Thus, a = − 2 and b = 1. This is indeed a solution,
   as can be seen by finishing the computation.
   ( ) ( )
   x x 6
   6 (2008)