微积分(A)随缘一题[13]

不能能洛必达，邻域不可导

(1)

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

(2)

\[\lim_{x \to 0} \frac{2^xf(x)-1}{x}=\lim_{x \to 0} \frac{2^x(f(x)-f(0))+2^x-1}{x}=\lim_{x \to 0} 2^xf'(0)+\frac{2^x-1}{x}=f'(0)+\ln 2=-1+\ln 2 \]

(3)

\[\lim_{x \to 1} \frac{f(\ln x)-1}{1-x}=\lim_{x \to 1} \frac{f(\ln x)-f(0)}{\ln x} \frac{\ln x}{1-x}=f'(0) \cdot (-1)=1 \]