第3章 3.25

(1)

$\lim_{x \to 0} \frac{\sin x - x}{x^3} = \lim_{x \to 0} \frac{\left(x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \cdots\right) - x}{x^3} = \lim_{x \to 0} \left(-\frac{1}{3!} + \frac{x^2}{5!} - \cdots\right) = -\frac{1}{6}$

(2)

$\lim_{x \to 0} \frac{e^x - 1 - x}{x^2} = \lim_{x \to 0} \frac{\left(1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots\right) - 1 - x}{x^2} = \lim_{x \to 0} \left(\frac{1}{2!} + \frac{x}{3!} + \cdots\right) = \frac{1}{2}$

(3)

$\lim_{x \to 0} \frac{\cos x + 2x - 1}{3x} = \lim_{x \to 0} \frac{\left(1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \cdots\right) + 2x - 1}{3x} = \lim_{x \to 0} \frac{2 - \dfrac{x}{2!} + \dfrac{x^3}{4!} - \cdots}{3} = \frac{2}{3}$

(4)

$\lim_{x \to 0} \frac{x\cos x - x}{\log(1+x^3)} = \lim_{x \to 0} \frac{x\left(1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \cdots\right) - x}{x^3 - \dfrac{x^6}{2} + \cdots} = \lim_{x \to 0} \frac{-\dfrac{1}{2!} + \dfrac{x^2}{4!} - \cdots}{1 - \dfrac{x^3}{2} + \cdots} = -\frac{1}{2}$