극한값과 미정계수

$\displaystyle \lim\limits _{x\rightarrow 1} ( x - 1 ) = 0$이므로 $\displaystyle \lim\limits _{x\rightarrow 1} ( x ^{2} + ax + b ) = 1 + a + b = 0$ $\therefore b = - 1 - a \cdots \text{㉠}$ 이 때, ㉠을 주어진 식에 대입하면 $\displaystyle \lim\limits _{x\rightarrow 1} \frac{x - 1}{x ^{2} + ax + b} = \lim\limits _{x\rightarrow 1} \frac{x - 1}{x ^{2} + ax - ( 1 + a )}$ $\displaystyle = \lim\limits _{x\rightarrow 1} \frac{x - 1}{( x - 1 ) ( x + 1 + a )}$ $\displaystyle = \lim\limits _{x\rightarrow 1} \frac{1}{( x + 1 + a )} = \frac{1}{a + 2}$$\displaystyle = \frac{1}{3}$ $\therefore a = 1$ 한편, ㉠에서 $b = - 2$ $\therefore a b = - 2$