limit as x approaches 0 of (12^{4x}-1)/(2x)

解

x \to 0 lim (\frac{1 2 ^{4 x} - 1}{2 x})

2 ln (12)

十進法表記

4.96981 \dots

解答ステップ

x \to 0 lim (\frac{1 2 ^{4 x} - 1}{2 x})

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= \frac{1}{2} \cdot x \to 0 lim (\frac{1 2 ^{4 x} - 1}{x})

1 2^{4 x} = 2073 6^{x}

= \frac{1}{2} \cdot x \to 0 lim (\frac{2073 6 ^{x} - 1}{x})

ロピタルの定理を適用する

= \frac{1}{2} \cdot x \to 0 lim (\frac{ln ( 12 ) \cdot 8 1 ^{x} \cdot 2 ^{8 x + 2}}{1})

値を挿入する

x = 0

= \frac{1}{2} \cdot \frac{ln ( 12 ) \cdot 8 1 ^{0} \cdot 2 ^{8 \cdot 0 + 2}}{1}

簡素化

\frac{1}{2} \cdot \frac{ln ( 12 ) \cdot 8 1 ^{0} \cdot 2 ^{8 \cdot 0 + 2}}{1} : 2 ln (12)

= 2 ln (12)