微积分(A)随缘一题[14]

(1)

\[\lim_{x \to 0}(1-2f(x))^{\frac{1}{\sin x}}=e^{-2\lim_{x \to 0}\frac{f(x)}{\sin x}}=e^{-2\lim_{x \to 0} \frac{f(x)-f(0)}{x-0} \cdot \frac{x}{\sin x}}=e^{-2f'(0)}=e^{-4} \]

(2)

\[\lim_{x \to 0} \frac{f(1-\cos x)}{\tan \sin x^2}=\lim_{x \to 0} \frac{\sin x^2}{\sin \sin x^2} \cdot \frac{f(1-\cos x)-f(0)}{1-\cos x} \cdot \frac{1- \cos x}{\frac{x^2}{2}} \cdot \frac{x^2}{\sin x^2} \cdot \frac{1}{2}=\frac{f'(0)}{2} \]

(3)

\[f(0)=0,f'(0)=1 \Rightarrow \lim_{x \to +\infty} x^{\frac{1}{2}}\sqrt{f(\frac{2}{x})}=\sqrt{\lim_{x \to \infty}\frac{f(\frac{2}{x})-f(0)}{\frac{2}{x}-0} \cdot 2}=\sqrt{2f'(0)}=\sqrt{2} \]