limit as x approaches 0+of 2/x-1/(e^x-1)

解

x \to 0 + lim (\frac{2}{x} - \frac{1}{e ^{x} - 1})

\infty

解答ステップ

x \to 0 + lim (\frac{2}{x} - \frac{1}{e ^{x} - 1})

\frac{2}{x} - \frac{1}{e ^{x} - 1} = \frac{2}{x} (1 - \frac{\frac{1}{e ^{x} - 1}}{\frac{2}{x}})

= x \to 0 + lim (\frac{2}{x} (1 - \frac{\frac{1}{e ^{x} - 1}}{\frac{2}{x}}))

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

= 2 \cdot x \to 0 + lim (\frac{1}{x} (1 - \frac{\frac{1}{e ^{x} - 1}}{\frac{2}{x}}))

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

= 2 \cdot x \to 0 + lim (\frac{1}{x}) \cdot x \to 0 + lim (1 - \frac{\frac{1}{e ^{x} - 1}}{\frac{2}{x}})

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

x \to 0 + lim (1 - \frac{\frac{1}{e ^{x} - 1}}{\frac{2}{x}}) = \frac{1}{2}

= 2 \cdot \infty \cdot \frac{1}{2}

無限大の特性を適用する:

c \cdot \infty = \infty

= \infty