극한값과 시그마의 계산

$\displaystyle \lim\limits _{x \rightarrow \infty} {\sum\limits _{k = 1} ^{4} \left\{ x f \left( 1 + \frac{3 ^{k}}{x} \right) g \left( 1 + \frac{3 ^{k}}{x} \right) \right.}$에서 $\displaystyle \frac{1}{x} = h$, $p \left( x \right) = f \left( x \right) g \left( x \right)$라 하면 $\displaystyle \lim\limits _{x \rightarrow \infty} {\sum\limits _{k = 1} ^{4} \left\{ x f \left( 1 + \frac{3 ^{k}}{x} \right) g \left( 1 + \frac{3 ^{k}}{x} \right) \right.}$ $\displaystyle = \lim\limits _{h \rightarrow 0} {\sum\limits _{k = 1} ^{4} \frac{p \left( 1 + 3 ^{k} h \right)}{h}}$ $\displaystyle = \lim\limits _{h \rightarrow 0} \sum\limits _{k = 1} ^{4} \frac{p \left( 1 + 3 ^{k} h \right) - p \left( 1 \right)}{3 ^{k} h} \cdot 3 ^{k}$ ($\because$$p \left( 1 \right) = f \left( 1 \right) g \left( 1 \right) = 0$) $\displaystyle = \sum\limits _{k = 1} ^{4} p' \left( 1 \right) \cdot 3 ^{k}$ $\displaystyle = \left\{ f' \left( 1 \right) g \left( 1 \right) + f \left( 1 \right) g' \left( 1 \right) \right\} \sum\limits _{k = 1} ^{4} 3 ^{k}$ $\displaystyle = 4 \sum\limits _{k = 1} ^{4} 3 ^{k}$ $\displaystyle = 4 \times \frac{3 \left( 3 ^{4} - 1 \right)}{3 - 1}$ $= 480$