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

解答

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

\frac{1}{2} ln (2)

十进制

0.34657 \dots

求解步骤

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

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

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

使用洛必达法则

= \frac{1}{2} \cdot x \to 0 lim (\frac{ln ( 2 ) \cdot 2 ^{x}}{1})

代入值

x = 0

= \frac{1}{2} \cdot \frac{ln ( 2 ) \cdot 2 ^{0}}{1}

化简

\frac{1}{2} \cdot \frac{ln ( 2 ) \cdot 2 ^{0}}{1} : \frac{1}{2} ln (2)

= \frac{1}{2} ln (2)

x \to 0 lim ((- 4 ln (x))^{x})

x \to 8 - lim (2 x - 8)

x \to 7 - lim (\frac{x - 8}{x - 7})

x \to 2 lim (2 x^{2} - 3 x - 4)

x \to 2 lim (\frac{4}{3} x - \frac{1}{2})