limit as x approaches 0 of (3^{sin(x)}-1)/(4x)

解

x \to 0 lim (\frac{3 ^{s i n (x)} - 1}{4 x})

\frac{1}{4} ln (3)

十進法表記

0.27465 \dots

解答ステップ

x \to 0 lim (\frac{3 ^{s i n (x)} - 1}{4 x})

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

= \frac{1}{4} \cdot x \to 0 lim (\frac{3 ^{s i n (x)} - 1}{x})

ロピタルの定理を適用する

= \frac{1}{4} \cdot x \to 0 lim (\frac{ln ( 3 ) \cdot 3 ^{s i n (x)} cos ( x )}{1})

値を挿入する

x = 0

= \frac{1}{4} \cdot \frac{ln ( 3 ) \cdot 3 ^{s i n (0)} cos ( 0 )}{1}

簡素化

\frac{1}{4} \cdot \frac{ln ( 3 ) \cdot 3 ^{s i n (0)} cos ( 0 )}{1} : \frac{1}{4} ln (3)

= \frac{1}{4} ln (3)