극한값이 0 조건

$f \left( 1 \right) = 0$에서 $f \left( x \right) = \left( x - 1 \right) \left( x - a \right) \left( a \right.$는 실수$\left. \right)$ (ⅰ) $a = 1$일 때 $\displaystyle \lim\limits _{x \rightarrow 1} \frac{f \left( x + 3 \right) \times g \left( x \right)}{\left\{ f \left( x \right) \right\} ^{2}} = \lim\limits _{x \rightarrow 1} \frac{\left( x + 2 \right) ^{2} g \left( x \right)}{\left( x - 1 \right) ^{4}}$ 사차다항식 $\left( x + 2 \right) ^{2} g \left( x \right)$는 $\left( x - 1 \right) ^{5}$을 인수로 가질 수 없으므로 $\displaystyle \lim\limits _{x \rightarrow 1} \frac{f \left( x + 3 \right) \times g \left( x \right)}{\left\{ f \left( x \right) \right\} ^{2}} = 0$을 만족시키지 않는다. (ⅱ) $a \neq 1$일 때 $\displaystyle \lim\limits _{x \rightarrow 1} \frac{f \left( x + 3 \right) \times g \left( x \right)}{\left\{ f \left( x \right) \right\} ^{2}}$ $\displaystyle = \lim\limits _{x \rightarrow 1} \frac{\left( x + 2 \right) \left( x + 3 - a \right) g \left( x \right)}{\left( x - 1 \right) ^{2} \left( x - a \right) ^{2}} = 0$ 이므로 사차다항식 $\left( x + 2 \right) \left( x + 3 - a \right) g \left( x \right)$는 $\left( x - 1 \right) ^{3}$을 인수로 갖는다.| 그러므로 $\left( x + 3 - a \right) g \left( x \right) = \left( x - 1 \right) ^{3}$이고, $a = 4$, $g \left( x \right) = \left( x - 1 \right) ^{2}$ (ⅰ), (ⅱ)에 의하여 $f \left( x \right) = \left( x - 1 \right) \left( x - 4 \right)$, $g \left( x \right) = \left( x - 1 \right) ^{2}$ 따라서 $f \left( 5 \right) + g \left( 5 \right) = 4 + 16 = 20$