limit as x approaches 0 of (cos(5x)tan(5x))/(3x)

解

x \to 0 lim (\frac{cos ( 5 x ) tan ( 5 x )}{3 x})

\frac{5}{3}

十進法表記

1.66666 \dots

解答ステップ

x \to 0 lim (\frac{cos ( 5 x ) tan ( 5 x )}{3 x})

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

= \frac{1}{3} \cdot x \to 0 lim (\frac{cos ( 5 x ) tan ( 5 x )}{x})

cos (5 x) tan (5 x) = sin (5 x)

= \frac{1}{3} \cdot x \to 0 lim (\frac{sin ( 5 x )}{x})

ロピタルの定理を適用する

= \frac{1}{3} \cdot x \to 0 lim (\frac{cos ( 5 x ) \cdot 5}{1})

値を挿入する

x = 0

= \frac{1}{3} \cdot \frac{cos ( 5 \cdot 0 ) \cdot 5}{1}

簡素化

\frac{1}{3} \cdot \frac{cos ( 5 \cdot 0 ) \cdot 5}{1} : \frac{5}{3}

= \frac{5}{3}