limit as x approaches infinity of (ln(x))/(e^x)

解

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

0

解答ステップ

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

ロピタルの定理を適用する

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

分数の規則を適用する:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

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

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

不定形式の場合を除く

= \frac{lim _{x \to \infty} ( 1 )}{lim _{x \to \infty} ( e ^{x} x )}

x \to \infty lim (1) = 1

x \to \infty lim (e^{x} x) = \infty

= \frac{1}{\infty}

無限大の特性を適用する:

\frac{c}{\infty} = 0

= 0