극한 조건 다항함수

$f ( x )$가 다항함수이고 $\displaystyle \lim\limits _{x \rightarrow \infty} {\frac{f ( x )}{x ^{2}} = 3}$이므로 $f ( x ) = 3 x ^{2} + ax + b$ ($a , b$는 실수) $\displaystyle \lim\limits _{x \rightarrow 2} {\frac{f ( x )}{x ^{2} - x - 2} = 6}$이고 $\displaystyle \lim\limits _{x \rightarrow 2} {\left( x ^{2} - x - 2 \right)} = 0$이므로 $\displaystyle \lim\limits _{x \rightarrow 2} {f ( x ) = f ( 2 ) = 0}$ $f ( 2 ) = 12 + 2 a + b = 0$, $b = - 2 a - 12$ $f ( x )$$= 3 x ^{2} + ax + ( - 2 a - 12 )$ $= ( x - 2 ) ( 3 x + 6 + a )$ $\displaystyle \lim\limits _{x \rightarrow 2} {\frac{f ( x )}{x ^{2} - x - 2}}$$\displaystyle = \lim\limits _{x \rightarrow 2} {\frac{( x - 2 ) ( 3 x + 6 + a )}{( x - 2 ) ( x + 1 )}}$ $\displaystyle = \frac{6 + 6 + a}{3}$$= 6$ $a = 6 , b = \mathit{-} 24$ $f ( x ) = 3 x ^{2} + 6 x - 24$ $f ( 0 ) = \mathit{-} 24$