Question

设函数f(x)=arctanx,若f(x)=xf′(ε)求limx→0ε²/x²

Answer 1

问：设函数f(x)=arctanx,若f(x)=xf′(ε)求limx→0ε²/x²

答：### 注意 1. $f'(x) = \frac{1}{1 + x^{2}}$ 2. $x \to 0$ 时，$\arctan x = x - \frac{1}{3}x^{3} + o(x^{3})$ 由于 $f(x) = x f'(\xi)$，所以可知 $f'(\xi) = \frac{1}{1 + \xi^{2}} = \frac{f(x)}{x} = \frac{\arctan x}{x}$，$\xi^{2} = \frac{x \cdot \arctan x}{\arctan x} = x$， $\lim_{x \to 0} \frac{\xi^{2}}{x^{2}} = \lim_{x \to 0} \frac{x \cdot \arctan x}{x^{2}\arctan x} = \lim_{x \to 0} \frac{x(x - \frac{1}{3}x^{3} + o(x^{3}))}{x^{3}} = \frac{1}{3}$。 故答案选：D。