limit as x approaches 0+of (x^x-1)/x

解

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

- \infty

解答ステップ

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

ロピタルの定理を適用する

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

簡素化

= x \to 0 + lim (x^{x} (ln (x) + 1))

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

不定形式の場合を除く

= x \to 0 + lim (x^{x}) \cdot x \to 0 + lim (ln (x) + 1)

x \to 0 + lim (x^{x}) = 1

x \to 0 + lim (ln (x) + 1) = - \infty

= 1 \cdot (- \infty)

無限大の特性を適用する:

c \cdot (- \infty) = - \infty

= - \infty