limit as x approaches 0 of (ln(e^x-x))/x

解

x \to 0 lim (\frac{ln ( e ^{x} - x )}{x})

0

解答ステップ

x \to 0 lim (\frac{ln ( e ^{x} - x )}{x})

\frac{ln ( e ^{x} - x )}{x} = ln (e^{x} - x) \frac{1}{x}

= x \to 0 lim (ln (e^{x} - x) \frac{1}{x})

簡素化

ln (e^{x} - x) \frac{1}{x} : \frac{ln ( e ^{x} - x )}{x}

= x \to 0 lim (\frac{ln ( e ^{x} - x )}{x})

ロピタルの定理を適用する

= x \to 0 lim (\frac{\frac{e ^{x} - 1}{e ^{x} - x}}{1})

値を挿入する

x = 0

= \frac{\frac{e ^{0} - 1}{e ^{0} - 0}}{1}

簡素化

\frac{\frac{e ^{0} - 1}{e ^{0} - 0}}{1} : 0

= 0